what is the meaning of probability experiment

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• Introduction

Applications of simple probability experiments

• The principle of additivity
• Multinomial probability
• The birthday problem
• Applications of conditional probability
• Independence
• Bayes’s theorem
• Random variables
• Probability distribution
• Expected value
• An alternative interpretation of probability
• The law of large numbers
• The central limit theorem
• The Poisson approximation
• Infinite sample spaces
• The strong law of large numbers
• Measure theory
• Probability density functions
• Conditional expectation and least squares prediction
• The Poisson process
• Brownian motion process
• Stationary processes
• Markovian processes
• The Ehrenfest model of diffusion
• The symmetric random walk
• Queuing models
• Insurance risk theory
• Martingale theory

• What was Carl Friedrich Gauss’s childhood like?
• What awards did Carl Friedrich Gauss win?
• How was Carl Friedrich Gauss influential?

probability theory

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• Stanford University - Review of Probability Theory
• Statistics LibreTexts - Probability Theory
• Indian Academy of Sciences - What is Probability Theory?
• University of California - Department of Statistics - Probability: Philosophy and Mathematical Background
• Stanford Encyclopedia of Philosophy - Quantum Logic and Probability Theory
• Table Of Contents

probability theory , a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance .

The word probability has several meanings in ordinary conversation. Two of these are particularly important for the development and applications of the mathematical theory of probability. One is the interpretation of probabilities as relative frequencies, for which simple games involving coins, cards, dice, and roulette wheels provide examples. The distinctive feature of games of chance is that the outcome of a given trial cannot be predicted with certainty, although the collective results of a large number of trials display some regularity. For example, the statement that the probability of “heads” in tossing a coin equals one-half, according to the relative frequency interpretation, implies that in a large number of tosses the relative frequency with which “heads” actually occurs will be approximately one-half, although it contains no implication concerning the outcome of any given toss. There are many similar examples involving groups of people, molecules of a gas, genes, and so on. Actuarial statements about the life expectancy for persons of a certain age describe the collective experience of a large number of individuals but do not purport to say what will happen to any particular person. Similarly, predictions about the chance of a genetic disease occurring in a child of parents having a known genetic makeup are statements about relative frequencies of occurrence in a large number of cases but are not predictions about a given individual.

(Read Steven Pinker’s Britannica entry on rationality.)

This article contains a description of the important mathematical concepts of probability theory, illustrated by some of the applications that have stimulated their development. For a fuller historical treatment, see probability and statistics . Since applications inevitably involve simplifying assumptions that focus on some features of a problem at the expense of others, it is advantageous to begin by thinking about simple experiments , such as tossing a coin or rolling dice, and later to see how these apparently frivolous investigations relate to important scientific questions.

Experiments, sample space, events, and equally likely probabilities

The fundamental ingredient of probability theory is an experiment that can be repeated, at least hypothetically, under essentially identical conditions and that may lead to different outcomes on different trials. The set of all possible outcomes of an experiment is called a “sample space.” The experiment of tossing a coin once results in a sample space with two possible outcomes, “heads” and “tails.” Tossing two dice has a sample space with 36 possible outcomes, each of which can be identified with an ordered pair ( i , j ), where i and j assume one of the values 1, 2, 3, 4, 5, 6 and denote the faces showing on the individual dice. It is important to think of the dice as identifiable (say by a difference in colour), so that the outcome (1, 2) is different from (2, 1). An “event” is a well-defined subset of the sample space. For example, the event “the sum of the faces showing on the two dice equals six” consists of the five outcomes (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1).

A third example is to draw n balls from an urn containing balls of various colours. A generic outcome to this experiment is an n -tuple, where the i th entry specifies the colour of the ball obtained on the i th draw ( i = 1, 2,…, n ). In spite of the simplicity of this experiment, a thorough understanding gives the theoretical basis for opinion polls and sample surveys. For example, individuals in a population favouring a particular candidate in an election may be identified with balls of a particular colour, those favouring a different candidate may be identified with a different colour, and so on. Probability theory provides the basis for learning about the contents of the urn from the sample of balls drawn from the urn; an application is to learn about the electoral preferences of a population on the basis of a sample drawn from that population.

Another application of simple urn models is to use clinical trials designed to determine whether a new treatment for a disease, a new drug, or a new surgical procedure is better than a standard treatment. In the simple case in which treatment can be regarded as either success or failure, the goal of the clinical trial is to discover whether the new treatment more frequently leads to success than does the standard treatment. Patients with the disease can be identified with balls in an urn. The red balls are those patients who are cured by the new treatment, and the black balls are those not cured. Usually there is a control group , who receive the standard treatment. They are represented by a second urn with a possibly different fraction of red balls. The goal of the experiment of drawing some number of balls from each urn is to discover on the basis of the sample which urn has the larger fraction of red balls. A variation of this idea can be used to test the efficacy of a new vaccine. Perhaps the largest and most famous example was the test of the Salk vaccine for poliomyelitis conducted in 1954. It was organized by the U.S. Public Health Service and involved almost two million children. Its success has led to the almost complete elimination of polio as a health problem in the industrialized parts of the world. Strictly speaking, these applications are problems of statistics , for which the foundations are provided by probability theory.

In contrast to the experiments described above, many experiments have infinitely many possible outcomes. For example, one can toss a coin until “heads” appears for the first time. The number of possible tosses is n = 1, 2,…. Another example is to twirl a spinner. For an idealized spinner made from a straight line segment having no width and pivoted at its centre, the set of possible outcomes is the set of all angles that the final position of the spinner makes with some fixed direction, equivalently all real numbers in [0, 2π). Many measurements in the natural and social sciences, such as volume, voltage, temperature, reaction time, marginal income, and so on, are made on continuous scales and at least in theory involve infinitely many possible values. If the repeated measurements on different subjects or at different times on the same subject can lead to different outcomes, probability theory is a possible tool to study this variability.

Because of their comparative simplicity, experiments with finite sample spaces are discussed first. In the early development of probability theory, mathematicians considered only those experiments for which it seemed reasonable, based on considerations of symmetry, to suppose that all outcomes of the experiment were “equally likely.” Then in a large number of trials all outcomes should occur with approximately the same frequency. The probability of an event is defined to be the ratio of the number of cases favourable to the event—i.e., the number of outcomes in the subset of the sample space defining the event—to the total number of cases. Thus, the 36 possible outcomes in the throw of two dice are assumed equally likely, and the probability of obtaining “six” is the number of favourable cases, 5, divided by 36, or 5/36.

Now suppose that a coin is tossed n times, and consider the probability of the event “heads does not occur” in the n tosses. An outcome of the experiment is an n -tuple, the k th entry of which identifies the result of the k th toss. Since there are two possible outcomes for each toss, the number of elements in the sample space is 2 n . Of these, only one outcome corresponds to having no heads, so the required probability is 1/2 n .

It is only slightly more difficult to determine the probability of “at most one head.” In addition to the single case in which no head occurs, there are n cases in which exactly one head occurs, because it can occur on the first, second,…, or n th toss. Hence, there are n + 1 cases favourable to obtaining at most one head, and the desired probability is ( n + 1)/2 n .

Experimental Probability

Experimental probability: introduction, experimental probability: definition, experimental probability formula, solved examples, practice problems, frequently asked questions.

In mathematics, probability refers to the chance of occurrence of a specific event. Probability can be measured on a scale from 0 to 1. The probability is 0 for an impossible event. The probability is 1 if the occurrence of the event is certain.

There are two approaches to study probability: experimental and theoretical. 

Suppose you and your friend toss a coin to decide who gets the first turn to ride a new bicycle. You choose “heads” and your friend chooses “tails.” 

Can you guess who will win? No! You have $\frac{1}{2}$ a chance of winning and so does your friend. This is theoretical since you are predicting the outcome based on what is expected to happen and not on the basis of outcomes of an experiment.

So, what is the experimental probability? Experimental probability is calculated by repeating an experiment and observing the outcomes. Let’s understand this a little better.

Experimental probability, or empirical probability, is the probability calculated by performing actual experiments and gathering or recording the necessary information. How would you define an experiment? The math definition of an experiment is “a process or procedure that can be repeated and that has a set of well-defined possible results or outcomes.”

Consider the same example. Suppose you flip the coin 50 times to see whether you get heads or tails, and you record the outcomes. Suppose you get heads 20 times and tails 30 times. Then the probability calculated using these outcomes is experimental probability. Here, t he experimental meaning is connected with such experiments used to determine the probability of an event.

Now that you know the meaning of experimental probability, let’s understand its formula.

Experimental Probability for an Event A can be calculated as follows:

P(E) $= \frac{Number of occurance of the event A}{Total number of trials}$

Let’s understand this with the help of the last example. 

A coin is flipped a total of 50 times. Heads appeared 20 times. Now, what is the experimental probability of getting heads?

E xperimental probability of getting heads $= \frac{Number of occurrences}{Total number of trials}$

P (Heads) $= \frac{20}{50} = \frac{2}{5}$

P (Tails) $= \frac{30}{50} = \frac{3}{5}$

Experimental Probability vs. Theoretical Probability

Theoretical probability expresses what is expected. On the other hand, experimental probability explains how frequently an event occurred in an experiment.

If you roll a die, the theoretical probability of getting any particular number, say 3, is $\frac{1}{6}$. 

However, if you roll the die 100 times and record how many times 3 appears on top, say 65 times, then the experimental probability of getting 3 is $\frac{65}{100}$.

Theoretical probability for Event A can be calculated as follows:

P(A) $= \frac{Number of outcomes favorable to Event A}{Number of possible outcomes}$

In the example of flipping a coin, the theoretical probability of the occurrence of heads (or tails) on tossing a coin is

P(H) $= \frac{1}{2}$ and  P(T) $= \frac{1}{2}$ (since possible outcomes are $2 -$ head or tail)

Experimental Probability: Examples

Let’s take a look at some of the examples of experimental probability .

Example 1: Ben tried to toss a ping-pong ball in a cup using 10 trials, out of which he succeeded 4 times. 

P(win) $= \frac{Number of success}{Number of trials}$

             $= \frac{4}{10}$

             $= \frac{2}{5}$

Example 2: Two students are playing a game of die. They want to know how many times they land on 2 on the dice if the die is rolled 20 times in a row. 

The experimental probability of rolling a 2 

$= \frac{Number of times 2 appeared}{Number of trials}$

$= \frac{5}{20}$

$= \frac{1}{4}$

1. Probability of an event always lies between 0 and 1.

2. You can also express the probability as a decimal and a percentage.

Experimental probability is a probability that is determined by the results of a series of experiments. Learn more such interesting concepts at SplashLearn .

1. Leo tosses a coin 25 times and observes that the “head” appears 10 times. What is the experimental probability of getting a head?

 P(Head) $= \frac{Number of times heads appeared}{Total number of trials}$

               $= \frac{10}{25}$

               $= \frac{2}{5}$

               $= 0.4$

2. The number of cakes a baker makes per day in a week is given as 7, 8, 6, 10, 2, 8, 3. What is the probability that the baker makes less than 6 cakes the next day?

Solution: 

Number of cakes baked each day in a week $= 7, 8, 6, 10, 2, 8, 3$

Out of 7 days, there were 2 days (highlighted in bold) on which the baker made less than 6 cookies.

P$(< 6 $cookies$) = \frac{2}{7}$

3. The chart below shows the number of times a number was shown on the face of a tossed die. What was the probability of getting a 3 in this experiment?

Number of times 3 showed $= 7$

Number of tosses $= 30$

P(3) $= \frac{7}{30}$

4. John kicked a ball 20 times. He kicked 16 field goals and missed 4 times . What is the experimental probability that John will kick a field goal during the game?

Solution:  

John succeeded in kicking 16 field goals. He attempted to kick a field goal 20 times. 

So, the number of trials $= 20$

John’s experimental probability of kicking a field goal $= \frac{Successful outcomes} {Trials attempted} = \frac{16}{20}$ 

$= \frac{4}{5}$

$= 0.8$ or $80%$

5. James recorded the color of bikes crossing his street. Of the 500 bikes, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were gray. What is the probability that the car crossing his street is white?

Number of white bikes $= 100$ 

Total number of bikes $= 500$

P(white bike) $=  \frac{100}{500} = \frac{1}{5}$

Attend this quiz & Test your knowledge.

In a class, a student is chosen randomly in five trials to participate in 5 different events. Out of chosen students, 3 were girls and 2 were boys. What is the experimental probability of choosing a boy in the next event?

A manufacturer makes 1000 tablets every month. after inspecting 100 tablets, the manufacturer found that 30 tablets were defective. what is the probability that you will buy a defective tablet, the 3 coins are tossed 1000 times simultaneously and we get three tails $= 160$, two tails $= 260$, one tail $= 320$, no tails $= 260$. what is the probability of occurrence of two tails, the table below shows the colors of shirts sold in a clothing store on a particular day and their respective frequencies. use the table to answer the questions that follow. what is the probability of selling a blue shirt.

Jason leaves for work at the same time each day. Over a period of 327 working days, on his way to work, he had to wait for a train at the railway crossing for 68 days. What is the experimental probability that Jason has to wait for a train on his way to work?

What is the importance of experimental probability?

Experimental probability is widely used in research and experiments in various fields, such as medicine, social sciences, investing, and weather forecasting.

Is experimental probability always accurate?

Predictions based on experimental probability are less reliable than those based on theoretical probability.

Can experimental probability change every time the experiment is performed?

Since the experimental probability is based on the actual results of an experiment, it can change when the results of an experiment change.

What is theoretical probability?

The theoretical probability is calculated by finding the ratio of the number of favorable outcomes to the total number of probable outcomes.

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What is experimental probability? 

Practice questions, experimental probability – explanation & examples.

Experimental probability is the probability determined based on the results from performing the particular experiment. 

In this lesson we will go through:

• The meaning of experimental probability
• How to find experimental probability

The ratio of the number of outcomes favorable to an event to the total number of trials of the experiment.

Experimental Probability can be expressed mathematically as: 

$P(\text{E}) = \frac{\text{number of outcomes favorable to an event}}{\text{total number of trials of the experiment}}$

Let’s go back to the die tossing example. If after 12 throws you get one 6, then the experimental probability is $\frac{1}{12}$.  You can compare that to the theoretical probability. The theoretical probability of getting a 6 is $\frac{1}{6}$. This means that in 12 throws we would have expected to get 6 twice. 

Similarly, if in those 12 tosses you got a 1 five times, the experimental probability is $\frac{5}{12}$. 

How do we find experimental probability?

Now that we understand what is meant by experimental probability, let’s go through how it is found. 

To find the experimental probability of an event, divide the number of observed outcomes favorable to the event by the total number of trials of the experiment. 

Let’s go through some examples. 

Example 1:  There are 20 students in a class. Each student simultaneously flipped one coin. 12 students got a Head. From this experiment, what was the experimental probability of getting a head?

Number of coins showing Heads: 12

Total number of coins flipped: 20

$P(\text{Head}) = \frac{12}{20} = \frac{3}{5}$ 

Example 2:  The tally chart below shows the number of times a number was shown on the face of a tossed die. 

a. What was the probability of a 3 in this experiment?

b. What was the probability of a prime number?

First, sum the numbers in the frequency column to see that the experiment was performed 30 times. Then find the probabilities of the specified events. 

a. Number of times 3 showed = 7

Number of tosses = 30

$P(\text{3}) = \frac{7}{30}$ 

b. Frequency of primes = 6 + 7 + 2 = 15

Number of trials = 30 

$P(\text{prime}) = \frac{15}{30} = \frac{1}{2}$

Experimental probability can be used to predict the outcomes of experiments. This is shown in the following examples. 

Example 3: The table shows the attendance schedule of an employee for the month of May.

a. What is the probability that the employee is absent? 

b. How many times would we expect the employee to be present in June?

a. The employee was absent three times and the number of days in this experiment was 31. Therefore:

$P(\text{Absent}) = \frac{3}{31}$

b.  We expect the employee to be absent

$\frac{3}{31} × 30 = 2.9 ≈ 3$ days in June 

Example 4:  Tommy observed the color of cars owned by people in his small hometown. Of the 500 cars in town, 10 were custom colors, 100 were white, 50 were red, 120 were black, 100 were silver, 60 were blue, and 60 were grey. 

a. What is the probability that a car is red?

b. If a new car is bought by someone in town, what color do you think it would be? Explain. 

a. Number of red cars = 50 

Total number of cars = 500 

$P(\text{red car}) = \frac{50}{500} = \frac{1}{10}$ 

b. Based on the information provided, it is most likely that the new car will be black. This is because it has the highest frequency and the highest experimental probability. 

Now it is time for you to try these examples. 

The table below shows the colors of jeans in a clothing store and their respective frequencies. Use the table to answer the questions that follow.

• What is the probability of selecting a brown jeans?
• What is the probability of selecting a blue or a white jeans?

On a given day, a fast food restaurant notices that it sold 110 beef burgers, 60 chicken sandwiches, and 30 turkey sandwiches. From this observation, what is the experimental probability that a customer buys a beef burger?

Over a span of 20 seasons, a talent competition notices the following. Singers won 12 seasons, dancers won 2 seasons, comedians won 3 seasons, a poet won 1 season, and daring acts won the other 2 seasons. 

a. What is the experimental probability of a comedian winning  a season?

b. From the next 10 seasons, how many winners do you expect to be dancers?

Try this at home! Flip a coin 10 times. Record the number of tails you get. What is your P(tail)?

Number of brown jeans = 25

Total Number of jeans = 125

$P(\text{brown}) = \frac{25}{125}  = \frac{1}{5}$

Number of jeans that are blue or white = 75 + 20 = 95

$P(\text{blue or white}) = \frac{95}{125} = \frac{19}{25}$

Number of beef burgers = 110 

Number of burgers (or sandwiches) sold = 200 

$P(\text{beef burger}) = \frac{110}{200} = \frac{11}{20}$ 

a. Number of comedian winners = 3

Number of seasons = 20 

$P(\text{comedian}) = \frac{3}{20}$ 

b. First find the experimental probability that the winner is a dancer. 

Number of winners that are dancers = 2 

$P(\text{dancer}) = \frac{2}{20} = \frac{1}{10}$ 

Therefore we expect 

$\frac{1}{10} × 10 = 1$ winner to be a dancer in the next 10 seasons.

To find your P(tail) in 10 trials, complete the following with the number of tails you got. 

$P(\text{tail}) = \frac{\text{number of tails}}{10}$ 

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Experimental probability

Experimental probability (EP), also called empirical probability or relative frequency , is probability based on data collected from repeated trials.

Experimental probability formula

Let n represent the total number of trials or the number of times an experiment is done. Let p represent the number of times an event occurred while performing this experiment n times.

Example #1: A manufacturer makes 50,000 cell phones every month. After inspecting 1000 phones, the manufacturer found that 20 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month.

The total number of times the experiment is conducted is n = 1000

The number of times an event occurred is p  = 20 

Experimental probability is performed when authorities want to know how the public feels about a matter. Since it is not possible to ask every single person in the country, they may conduct a survey by asking a sample of the entire population. This is called population sampling. Example #2 is an example of this situation.

There are about 319 million people living in the USA. Pretend that a survey of 1 million people revealed that 300,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like electric car? How many people like electric cars?

Notice that the number of people who do not like electric cars is 1000000 - 300000 = 700000

Difference between experimental probability and theoretical probability

You can argue the same thing using a die, a coin, and a spinner. We will though use a coin and a spinner to help you see the difference.

Using a coin 

In theoretical probability, we say that "each outcome is equally likely " without the actual experiment. For instance, without  flipping a coin, you know that the outcome could either be heads or tails.  If the coin is not altered, we argue that each outcome (heads or tails) is equally likely. In other words, we are saying that in theory or (supposition, conjecture, speculation, assumption, educated guess) the probability to get heads is 50% or the probability to get tails in 50%. Since you did not actually flip the coin, you are making an assumption based on logic.

The logic is that there are 2 possible outcomes and since you are choosing 1 of the 2 outcomes, the probability is 1/2 or 50%. This is theoretical probability or guessing probability or probability based on assumption.

In the example above about flipping a coin, suppose you are looking for the probability to get a head. 

Then, the number of favorable outcomes is 1 and the number of possible outcomes is 2.

In experimental probability,  we want to take the guess work out of the picture, by doing the experiment to see how many times heads or teals will come up. If you flip a coin 1000 times, you might realize that it landed on heads only 400 times. In this case, the probability to get heads is only 40%. 

Your experiment may not even show tails until after the 4th flip and yet in the end you ended up with more tails than heads. 

If you repeat the experiment another day, you may find a completely different result. May be this time the number of heads is 600 and the number of tails is 400.

Using a spinner

Suppose a spinner has four equal-sized sections that are red, green, black, and yellow. 

In theoretical probability, you will not spin the spinner. Instead, you will say that the probability to get green is one-fourth or 25%. Why 25%? The total number of outcomes is 4 and the number of favorable outcomes is 1.

1/4 = 0.25 = 25%

However, in experimental probability, you may decide to spin the spinner 50 times or even more to see how many times you will get each color.

Suppose you spin the spinner 50 times. It is quite possible that you may end up with the result shown below:

Red: 10 Green: 15 Black: 5 Yellow: 20

Now, the probability to get green is 15/50 = 0.3 = 30%

As you can see, experimental probability is based more on facts, data collected, experiment or research!

Theoretical probability

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7.6: Basic Concepts of Probability

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Learning Objectives

After completing this section, you should be able to:

• Define probability including impossible and certain events.
• Calculate basic theoretical probabilities.
• Calculate basic empirical probabilities.
• Distinguish among theoretical, empirical, and subjective probability.
• Calculate the probability of the complement of an event.

It all comes down to this. The game of Monopoly that started hours ago is in the home stretch. Your sister has the dice, and if she rolls a 4, 5, or 7 she’ll land on one of your best spaces and the game will be over. How likely is it that the game will end on the next turn? Is it more likely than not? How can we measure that likelihood? This section addresses this question by introducing a way to measure uncertainty.

Introducing Probability

Uncertainty is, almost by definition, a nebulous concept. In order to put enough constraints on it that we can mathematically study it, we will focus on uncertainty strictly in the context of experiments. Recall that experiments are processes whose outcomes are unknown; the sample space for the experiment is the collection of all those possible outcomes. When we want to talk about the likelihood of particular outcomes, we sometimes group outcomes together; for example, in the Monopoly example at the beginning of this section, we were interested in the roll of 2 dice that might fall as a 4, 5, or 7. A grouping of outcomes that we’re interested in is called an event . In other words, an event is a subset of the sample space of an experiment; it often consists of the outcomes of interest to the experimenter.

Once we have defined the event that interests us, we can try to assess the likelihood of that event. We do that by assigning a number to each event ( E E ) called the probability of that event ( P ( E ) P ( E ) ). The probability of an event is a number between 0 and 1 (inclusive). If the probability of an event is 0, then the event is impossible. On the other hand, an event with probability 1 is certain to occur. In general, the higher the probability of an event, the more likely it is that the event will occur.

Example 7.16

Determining certain and impossible events.

Consider an experiment that consists of rolling a single standard 6-sided die (with faces numbered 1-6). Decide if these probabilities are equal to zero, equal to one, or somewhere in between.

• P ( roll a 4 ) P ( roll a 4 )
• P ( roll a 7 ) P ( roll a 7 )
• P ( roll a positive number ) P ( roll a positive number )
• P ( roll a 1 3 ) P ( roll a 1 3 )
• P ( roll an even number ) P ( roll an even number )
• P ( roll a single-digit number ) P ( roll a single-digit number )

Let's start by identifying the sample space. For one roll of this die, the possible outcomes are {1, 2, 3, 4, 5,6}. We can use that to assess these probabilities:

• We see that 4 is in the sample space, so it’s possible that it will be the outcome. It’s not certain to be the outcome, though. So, 0 < P ( roll a 4 ) < 1 0 < P ( roll a 4 ) < 1 .
• Notice that 7 is not in the sample space. So, P ( roll a 7 ) = 0 P ( roll a 7 ) = 0 .
• Every outcome in the sample space is a positive number, so this event is certain. Thus, P ( roll a positive number ) = 1 P ( roll a positive number ) = 1 .
• Since 1 3 1 3 is not in the sample space, P ( roll a 1 3 ) = 0 P ( roll a 1 3 ) = 0 .
• Some outcomes in the sample space are even numbers (2, 4, and 6), but the others aren’t. So, 0 < P ( roll an even number ) < 1 0 < P ( roll an even number ) < 1 .
• Every outcome in the sample space is a single-digit number, so P ( roll a single-digit number ) = 1 P ( roll a single-digit number ) = 1 .

Your Turn 7.16

Three ways to assign probabilities.

The probabilities of events that are certain or impossible are easy to assign; they’re just 1 or 0, respectively. What do we do about those in-between cases, for events that might or might not occur? There are three methods to assign probabilities that we can choose from. We’ll discuss them here, in order of reliability.

Method 1: Theoretical Probability

The theoretical method gives the most reliable results, but it cannot always be used. If the sample space of an experiment consists of equally likely outcomes, then the theoretical probability of an event is defined to be the ratio of the number of outcomes in the event to the number of outcomes in the sample space.

For an experiment whose sample space S S consists of equally likely outcomes, the theoretical probability of the event E E is the ratio

P ( E ) = n ( E ) n ( S ) , P ( E ) = n ( E ) n ( S ) ,

where n ( E ) n ( E ) and n ( S ) n ( S ) denote the number of outcomes in the event and in the sample space, respectively.

Example 7.17

Computing theoretical probabilities.

Recall that a standard deck of cards consists of 52 unique cards which are labeled with a rank (the whole numbers from 2 to 10, plus J, Q, K, and A) and a suit ( ♣ ♣ , ♢ ♢ , ♡ ♡ , or ♠ ♠ ). A standard deck is thoroughly shuffled, and you draw one card at random (so every card has an equal chance of being drawn). Find the theoretical probability of each of these events:

• The card is 10 ♠ 10 ♠ .
• The card is a ♡ ♡ .
• The card is a king (K).

There are 52 cards in the deck, so the sample space for each of these experiments has 52 elements. That will be the denominator for each of our probabilities.

• There is only one 10 ♠ 10 ♠ in the deck, so this event only has one outcome in it. Thus, P ( 10 ♠ ) = 1 52 P ( 10 ♠ ) = 1 52 .
• There are 13 ♡ s ♡ s in the deck, so P ( ♡ ) = 13 52 = 1 4 P ( ♡ ) = 13 52 = 1 4 .
• There are 4 cards of each rank in the deck, so P ( K ) = 4 52 = 1 13 P ( K ) = 4 52 = 1 13 .

Your Turn 7.17

It is critical that you make sure that every outcome in a sample space is equally likely before you compute theoretical probabilities!

Example 7.18

Using tables to find theoretical probabilities.

In the Basic Concepts of Probability, we were considering a Monopoly game where, if your sister rolled a sum of 4, 5, or 7 with 2 standard dice, you would win the game. What is the probability of this event? Use tables to determine your answer.

We should think of this experiment as occurring in two stages: (1) one die roll, then (2) another die roll. Even though these two stages will usually occur simultaneously in practice, since they’re independent, it’s okay to treat them separately.

Step 1: Since we have two independent stages, let’s create a table (Figure 7.27), which is probably the most efficient method for determining the sample space.

Now, each of the 36 ordered pairs in the table represent an equally likely outcome.

Step 2: To make our analysis easier, let’s replace each ordered pair with the sum (Figure 7.28).

Step 3: Since the event we’re interested in is the one consisting of rolls of 4, 5, or 7. Let’s shade those in (Figure 7.29).

Our event contains 13 outcomes, so the probability that your sister rolls a losing number is 13 36 13 36 .

Your Turn 7.18

Example 7.19, using tree diagrams to compute theoretical probability.

If you flip a fair coin 3 times, what is the probability of each event? Use a tree diagram to determine your answer

• You flip exactly 2 heads.
• You flip 2 consecutive heads at some point in the 3 flips.
• All 3 flips show the same result.

Let’s build a tree to identify the sample space (Figure 7.30).

The sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}, which has 8 elements.

• Flipping exactly 2 heads occurs three times (HHT, HTH, THH), so the probability is 3 8 3 8 .
• Flipping 2 consecutive heads at some point in the experiment happens 3 times: HHH, HHT, THH. So, the probability is 3 8 3 8 .
• There are 2 outcomes that all show the same result: HHH and TTT. So, the probability is 2 8 = 1 4 2 8 = 1 4 .

Your Turn 7.19

People in mathematics, gerolamo cardano.

The first known text that provided a systematic approach to probabilities was written in 1564 by Gerolamo Cardano (1501–1576). Cardano was a physician whose illegitimate birth closed many doors that would have otherwise been open to someone with a medical degree in 16th-century Italy. As a result, Cardano often turned to gambling to help ends meet. He was a remarkable mathematician, and he used his knowledge to gain an edge when playing at cards or dice. His 1564 work, titled Liber de ludo aleae (which translates as Book on Games of Chance ), summarized everything he knew about probability. Of course, if that book fell into the hands of those he played against, his advantage would disappear. That’s why he never allowed it to be published in his lifetime (it was eventually published in 1663). Cardano made other contributions to mathematics; he was the first person to publish the third degree analogue of the Quadratic Formula (though he didn’t discover it himself), and he popularized the use of negative numbers.

Method 2: Empirical Probability

Theoretical probabilities are precise, but they can’t be found in every situation. If the outcomes in the sample space are not equally likely, then we’re out of luck. Suppose you’re watching a baseball game, and your favorite player is about to step up to the plate. What is the probability that he will get a hit?

In this case, the sample space is {hit, not a hit}. That doesn’t mean that the probability of a hit is 1 2 1 2 , since those outcomes aren’t equally likely. The theoretical method simply can’t be used in this situation. Instead, we might look at the player’s statistics up to this point in the season, and see that he has 122 hits in 531 opportunities. So, we might think that the probability of a hit in the next plate appearance would be about 122 531 ≈ 0.23 122 531 ≈ 0.23 . When we use the outcomes of previous replications of an experiment to assign a probability to the next replication, we’re defining an empirical probability . Empirical probability is assigned using the outcomes of previous replications of an experiment by finding the ratio of the number of times in the previous replications the event occurred to the total number of previous replications.

Empirical probabilities aren’t exact, but when the number of previous replications is large, we expect them to be close. Also, if the previous runs of the experiment are not conducted under the exact set of circumstances as the one we’re interested in, the empirical probability is less reliable. For instance, in the case of our favorite baseball player, we might try to get a better estimate of the probability of a hit by looking only at his history against left- or right-handed pitchers (depending on the handedness of the pitcher he’s about to face).

Probability and Statistics

One of the broad uses of statistics is called statistical inference, where statisticians use collected data to make a guess (or inference) about the population the data were collected from. Nearly every tool that statisticians use for inference is based on probability. Not only is the method we just described for finding empirical probabilities one type of statistical inference, but some more advanced techniques in the field will give us an idea of how close that empirical probability might be to the actual probability!

Example 7.20

Finding empirical probabilities.

Assign an empirical probability to the following events:

• Jose is on the basketball court practicing his shots from the free throw line. He made 47 out of his last 80 attempts. What is the probability he makes his next shot?
• Amy is about to begin her morning commute. Over her last 60 commutes, she arrived at work 12 times in under half an hour. What is the probability that she arrives at work in 30 minutes or less?
• Felix is playing Yahtzee with his sister. Felix won 14 of the last 20 games he played against her. How likely is he to win this game?
• Since Jose made 47 out of his last 80 attempts, assign this event an empirical probability of 47 80 ≈ 59 % 47 80 ≈ 59 % .
• Amy completed the commute in under 30 minutes in 12 of the last 60 commutes, so we can estimate her probability of making it in under 30 minutes this time at 12 60 = 20 % 12 60 = 20 % .
• Since Felix has won 14 of the last 20 games, assign a probability for a win this time of 14 20 = 70 % 14 20 = 70 % .

Your Turn 7.20

Work it out, buffon’s needle.

A famous early question about probability (posed by Georges-Louis Leclerc, Comte de Buffon in the 18th century) had to do with the probability that a needle dropped on a floor finished with wooden slats would lay across one of the seams. If the distance between the slats is exactly the same length as the needle, then it can be shown using calculus that the probability that the needle crosses a seam is 2 π 2 π . Using toothpicks or matchsticks (or other uniformly long and narrow objects), assign an empirical probability to this experiment by drawing parallel lines on a large sheet of paper where the distance between the lines is equal to the length of your dropping object, then repeatedly dropping the objects and noting whether the object touches one of the lines. Once you have your empirical probability, take its reciprocal and multiply by 2. Is the result close to π π ?

Method 3: Subjective Probability

In cases where theoretical probability can’t be used and we don’t have prior experience to inform an empirical probability, we’re left with one option: using our instincts to guess at a subjective probability . A subjective probability is an assignment of a probability to an event using only one’s instincts.

Subjective probabilities are used in cases where an experiment can only be run once, or it hasn’t been run before. Because subjective probabilities may vary widely from person to person and they’re not based on any mathematical theory, we won’t give any examples. However, it’s important that we be able to identify a subjective probability when we see it; they will in general be far less accurate than empirical or theoretical probabilities.

Example 7.21

Distinguishing among theoretical, empirical, and subjective probabilities.

Classify each of the following probabilities as theoretical, empirical, or subjective.

• An eccentric billionaire is testing a brand new rocket system. He says there is a 15% chance of failure.
• With 4 seconds to go in a close basketball playoff game, the home team need 3 points to tie up the game and send it to overtime. A TV commentator says that team captain should take the final 3-point shot, because he has a 38% chance of making it (greater than every other player on the team).
• Felix is losing his Yahtzee game against his sister. He has one more chance to roll 2 dice; he’ll win the game if they both come up 4. The probability of this is about 2.8%.
• This experiment has never been run before, so the given probability is subjective.
• Presumably, the commentator has access to each player’s performance statistics over the entire season. So, the given probability is likely empirical.
• Rolling 2 dice results in a sample space with equally likely outcomes. This probability is theoretical. (We’ll learn how to calculate that probability later in this chapter.)

Your Turn 7.21

Benford’s law.

In 1938, Frank Benford published a paper (“The law of anomalous numbers,” in Proceedings of the American Philosophical Society ) with a surprising result about probabilities. If you have a list of numbers that spans at least a couple of orders of magnitude (meaning that if you divide the largest by the smallest, the result is at least 100), then the digits 1–9 are not equally likely to appear as the first digit of those numbers, as you might expect. Benford arrived at this conclusion using empirical probabilities; he found that 1 was about 6 times as likely to be the initial digit as 9 was!

New Probabilities from Old: Complements

One of the goals of the rest of this chapter is learning how to break down complicated probability calculations into easier probability calculations. We’ll look at the first of the tools we can use to accomplish this goal in this section; the rest will come later.

Given an event E E , the complement of E E (denoted E ′ E ′ ) is the collection of all of the outcomes that are not in E E . (This is language that is taken from set theory, which you can learn more about elsewhere in this text.) Since every outcome in the sample space either is or is not in E E , it follows that n ( E ) + n ( E ′ ) = n ( S ) n ( E ) + n ( E ′ ) = n ( S ) . So, if the outcomes in S S are equally likely, we can compute theoretical probabilities P ( E ) = n ( E ) n ( S ) P ( E ) = n ( E ) n ( S ) and P ( E ′ ) = n ( E ′ ) n ( S ) P ( E ′ ) = n ( E ′ ) n ( S ) . Then, adding these last two equations, we get

P ( E ) + P ( E ′ ) = n ( E ) n ( S ) + n ( E ′ ) n ( S ) = n ( E ) + n ( E ′ ) n ( S ) = n ( S ) n ( S ) = 1 P ( E ) + P ( E ′ ) = n ( E ) n ( S ) + n ( E ′ ) n ( S ) = n ( E ) + n ( E ′ ) n ( S ) = n ( S ) n ( S ) = 1

Thus, if we subtract P ( E ′ ) P ( E ′ ) from both sides, we can conclude that P ( E ) = 1 − P ( E ′ ) P ( E ) = 1 − P ( E ′ ) . Though we performed this calculation under the assumption that the outcomes in S S are all equally likely, the last equation is true in every situation.

P ( E ) = 1 − P ( E ′ ) P ( E ) = 1 − P ( E ′ )

How is this helpful? Sometimes it is easier to compute the probability that an event won’t happen than it is to compute the probability that it will . To apply this principle, it’s helpful to review some tricks for dealing with inequalities. If an event is defined in terms of an inequality, the complement will be defined in terms of the opposite inequality: Both the direction and the inclusivity will be reversed, as shown in the table below.

Example 7.22

Using the formula for complements to compute probabilities.

• If you roll a standard 6-sided die, what is the probability that the result will be a number greater than one?
• If you roll two standard 6-sided dice, what is the probability that the sum will be 10 or less?
• If you flip a fair coin 3 times, what is the probability that at least one flip will come up tails?
• Here, the sample space is {1, 2, 3, 4, 5, 6}. It’s easy enough to see that the probability in question is 5 6 5 6 , because there are 5 outcomes that fall into the event “roll a number greater than 1.” Let’s also apply our new formula to find that probability. Since E E is defined using the inequality roll > 1 roll > 1 , then E ′ E ′ is defined using roll ≤ 1 roll ≤ 1 . Since there’s only one outcome (1) in E ′ E ′ , we have P ( E ′ ) = 1 6 P ( E ′ ) = 1 6 . Thus, P ( E ) = 1 − P ( E ′ ) = 5 6 P ( E ) = 1 − P ( E ′ ) = 5 6 .

Here, the event E E is defined by the inequality sum ≤ 10 sum ≤ 10 . Thus, E ′ E ′ is defined by sum > 10 sum > 10 . There are three outcomes in E ′ E ′ : two 11s and one 12. Thus, P ( E ) = 1 − P ( E ′ ) = 1 − 3 36 = 11 12 P ( E ) = 1 − P ( E ′ ) = 1 − 3 36 = 11 12 .

• In Example 7.15, we found the sample space for this experiment consisted of these equally likely outcomes: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Our event E E is defined by T ≥ 1 T ≥ 1 , so E ′ E ′ is defined by T < 1 T < 1 . The only outcome in E ′ E ′ is the first one on the list, where zero tails are flipped. So, P ( E ) = 1 − P ( E ′ ) = 1 − 1 8 = 7 8 P ( E ) = 1 − P ( E ′ ) = 1 − 1 8 = 7 8 .

Your Turn 7.22

Check your understanding, section 7.5 exercises.

For the following exercises, use the following table of the top 15 players by number of plate appearances (PA) in the 2019 Major League Baseball season to assign empirical probabilities to the given events. A plate appearance is a batter’s opportunity to try to get a hit. The other columns are runs scored (R), hits (H), doubles (2B), triples (3B), home runs (HR), walks (BB), and strike outs (SO).

Probability

Probability defines the likelihood of occurrence of an event. There are many real-life situations in which we may have to predict the outcome of an event. We may be sure or not sure of the results of an event. In such cases, we say that there is a probability of this event to occur or not occur. Probability generally has great applications in games, in business to make predictions, and also it has extensive applications in this new area of artificial intelligence.

The probability of an event can be calculated by the probability formula by simply dividing the favourable number of outcomes by the total number of possible outcomes. The value of the probability of an event happening can lie between 0 and 1 because the favourable number of outcomes can never be more than the total number of outcomes. Also, the favorable number of outcomes cannot be negative. Let us discuss the basics of probability in detail in the following sections.

What is Probability?

Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event. For an experiment having 'n' number of outcomes, the number of favorable outcomes can be denoted by x. The formula to calculate the probability of an event is as follows.

Probability(Event) = Favorable Outcomes/Total Outcomes = x/n

Probability is used to predict the outcomes for the tossing of coins, rolling of dice, or drawing a card from a pack of playing cards. The probability is classified into two types:

• Theoretical probability
• Experimental probability

To understand each of these types, click on the respective links.

Terminology of Probability Theory

The following terms in probability theorey help in a better understanding of the concepts of probability.

Experiment: A trial or an operation conducted to produce an outcome is called an experiment.

Sample Space: All the possible outcomes of an experiment together constitute a sample space . For example, the sample space of tossing a coin is {head, tail}.

Favorable Outcome: An event that has produced the desired result or expected event is called a favorable outcome. For example, when we roll two dice, the possible/favorable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).

Trial: A trial denotes doing a random experiment.

Random Experiment: An experiment that has a well-defined set of outcomes is called a random experiment . For example, when we toss a coin, we know that we would get ahead or tail, but we are not sure which one will appear.

Event: The total number of outcomes of a random experiment is called an event .

Equally Likely Events: Events that have the same chances or probability of occurring are called equally likely events. The outcome of one event is independent of the other. For example, when we toss a coin, there are equal chances of getting a head or a tail.

Exhaustive Events: When the set of all outcomes of an event is equal to the sample space, we call it an exhaustive event .

Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events . For example, the climate can be either hot or cold. We cannot experience the same weather simultaneously.

Events in Probability

In probability theory, an event is a set of outcomes of an experiment or a subset of the sample space. If P(E) represents the probability of an event E, then, we have,

• P(E) = 0 if and only if E is an impossible event.
• P(E) = 1 if and only if E is a certain event.
• 0 ≤ P(E) ≤ 1.

Suppose, we are given two events, "A" and "B", then the probability of event A, P(A) > P(B) if and only if event "A" is more likely to occur than the event "B". Sample space(S) is the set of all of the possible outcomes of an experiment and n(S) represents the number of outcomes in the sample space.

P(E) = n(E)/n(S)

P(E’) = (n(S) - n(E))/n(S) = 1 - (n(E)/n(S))

E’ represents that the event will not occur.

Therefore, now we can also conclude that, P(E) + P(E’) = 1

Probability Formula

The probability equation defines the likelihood of the happening of an event. It is the ratio of favorable outcomes to the total favorable outcomes. The probability formula can be expressed as,

i.e., P(A) = n(A)/n(S)

• P(A) is the probability of an event 'B'.
• n(A) is the number of favorable outcomes of an event 'B'.
• n(S) is the total number of events occurring in a sample space.

Different Probability Formulas

Probability formula with addition rule : Whenever an event is the union of two other events, say A and B, then P(A or B) = P(A) + P(B) - P(A∩B) P(A ∪ B) = P(A) + P(B) - P(A∩B)

Probability formula with the complementary rule: Whenever an event is the complement of another event, specifically, if A is an event, then P(not A) = 1 - P(A) or P(A') = 1 - P(A). P(A) + P(A′) = 1.

Probability formula with the conditional rule : When event A is already known to have occurred, the probability of event B is known as conditional probability and is given by: P(B∣A) = P(A∩B)/P(A)

Probability formula with multiplication rule : Whenever an event is the intersection of two other events, that is, events A and B need to occur simultaneously. Then

• P(A ∩ B) = P(A)⋅P(B) (in case of independent events )
• P(A∩B) = P(A)⋅P(B∣A) (in case of dependent events )

Calculating Probability

In an experiment, the probability of an event is the possibility of that event occurring. The probability of any event is a value between (and including) "0" and "1". Follow the steps below for calculating probability of an event A:

• Step 1: Find the sample space of the experiment and count the elements. Denote it by n(S).
• Step 2: Find the number of favorable outcomes and denote it by n(A).
• Step 3: To find probability, divide n(A) by n(S). i.e., P(A) = n(A)/n(S).

Here are some examples that well describe the process of finding probability.

Example 1 : Find the probability of getting a number less than 5 when a dice is rolled by using the probability formula.

To find: Probability of getting a number less than 5 Given: Sample space, S = {1,2,3,4,5,6} Therefore, n(S) = 6

Let A be the event of getting a number less than 5. Then A = {1,2,3,4} So, n(A) = 4

Using the probability equation, P(A) = (n(A))/(n(s)) p(A) = 4/6 m = 2/3

Answer: The probability of getting a number less than 5 is 2/3.

Example 2: What is the probability of getting a sum of 9 when two dice are thrown?

There is a total of 36 possibilities when we throw two dice. To get the desired outcome i.e., 9, we can have the following favorable outcomes. (4,5),(5,4),(6,3)(3,6). There are 4 favorable outcomes. Probability of an event P(E) = (Number of favorable outcomes) ÷ (Total outcomes in a sample space) Probability of getting number 9 = 4 ÷ 36 = 1/9

Answer: Therefore the probability of getting a sum of 9 is 1/9.

Probability Tree Diagram

A tree diagram in probability is a visual representation that helps in finding the possible outcomes or the probability of any event occurring or not occurring. The tree diagram for the toss of a coin given below helps in understanding the possible outcomes when a coin is tossed. Each branch of the tree is associated with the respective probability (just like how 0.5 is written on each brack in the figure below). Remember that the sum of probabilities of all branches that start from the same point is always 1 (here, 0.5 + 0.5 = 1).

Types of Probability

There can be different perspectives or types of probabilities based on the nature of the outcome or the approach followed while finding probability of an event happening. The four types of probabilities are,

Classical Probability

Empirical probability, subjective probability, axiomatic probability.

Classical probability, often referred to as the "priori" or "theoretical probability", states that in an experiment where there are B equally likely outcomes, and event X has exactly A of these outcomes, then the probability of X is A/B, or P(X) = A/B. For example, when a fair die is rolled, there are six possible outcomes that are equally likely. That means, there is a 1/6 probability of rolling each number on the die.

The empirical probability or the experimental perspective evaluates probability through thought experiments. For example, if a weighted die is rolled, such that we don't know which side has the weight, then we can get an idea for the probability of each outcome by rolling the die number of times and calculating the proportion of times the die gives that outcome and thus find the probability of that outcome.

Subjective probability considers an individual's own belief of an event occurring. For example, the probability of a particular team winning a football match on a fan's opinion is more dependent upon their own belief and feeling and not on a formal mathematical calculation.

In axiomatic probability, a set of rules or axioms by Kolmogorov are applied to all the types. The chances of occurrence or non-occurrence of any event can be quantified by the applications of these axioms, given as,

• The smallest possible probability is zero, and the largest is one.
• An event that is certain has a probability equal to one.
• Any two mutually exclusive events cannot occur simultaneously, while the union of events says only one of them can occur.

Coin Toss Probability

Let us now look into the probability of tossing a coin . Quite often in games like cricket, for making a decision as to who would bowl or bat first, we sometimes use the tossing of a coin and decide based on the outcome of the toss. Let us check how we can use the concept of probability in the tossing of a single coin. Further, we shall also look into the tossing of two and three coins.

Tossing a Coin

A single coin on tossing has two outcomes, a head, and a tail. The concept of probability which is the ratio of favorable outcomes to the total number of outcomes can be used in finding probability of getting the head and the probability of getting a tail.

Total number of possible outcomes = 2; Sample Space = {H, T}; H: Head, T: Tail

• P(H) = Number of heads/Total outcomes = 1/2
• P(T)= Number of Tails/ Total outcomes = 1/2

Tossing Two Coins

In the process of tossing two coins, we have a total of four (= 2 2 ) outcomes. The probability formula can be used to find the probability of two heads, one head, no head, and a similar probability can be calculated for the number of tails. The probability calculations for the two heads are as follows.

Total number of outcomes = 4; Sample Space = {(H, H), (H, T), (T, H), (T, T)}

• P(2H) = P(0 T) = Number of outcome with two heads/Total Outcomes = 1/4
• P(1H) = P(1T) = Number of outcomes with only one head/Total Outcomes = 2/4 = 1/2
• P(0H) = (2T) = Number of outcome with two heads/Total Outcomes = 1/4

Tossing Three Coins

The number of total outcomes on tossing three coins simultaneously is equal to 2 3 = 8. For these outcomes, we can find the probability of getting one head, two heads, three heads, and no head. A similar probability can also be calculated for the number of tails.

Total number of outcomes = 2 3 = 8 Sample Space = {(H, H, H), (H, H, T), (H, T, H), (T, H, H), (T, T, H), (T, H, T), (H, T, T), (T, T, T)}

• P(0H) = P(3T) = Number of outcomes with no heads/Total Outcomes = 1/8
• P(1H) = P(2T) = Number of Outcomes with one head/Total Outcomes = 3/8
• P(2H) = P(1T) = Number of outcomes with two heads /Total Outcomes = 3/8
• P(3H) = P(0T) = Number of outcomes with three heads/Total Outcomes = 1/8

Dice Roll Probability

Many games use dice to decide the moves of players across the games. A dice has six possible outcomes and the outcomes of a dice is a game of chance and can be obtained by using the concepts of probability. Some games also use two dice, and there are numerous probabilities that can be calculated for outcomes using two dice. Let us now check the outcomes, their probabilities for one dice and two dice respectively.

Rolling One Dice

The total number of outcomes on rolling a die is 6, and the sample space is {1, 2, 3, 4, 5, 6}. Here we shall compute the following few probabilities to help in better understanding the concept of probability on rolling one dice.

• P( Even Number ) = Number of even number outcomes/Total Outcomes = 3/6 = 1/2
• P( Odd Number ) = Number of odd number outcomes/Total Outcomes = 3/6 = 1/2
• P( Prime Number ) = Number of prime number outcomes/Total Outcomes = 3/6 = 1/2

Rolling Two Dice

The total number of outcomes on rolling two dice is 6 2 = 36. The following image shows the sample space of 36 outcomes on rolling two dice.

Let us check a few probabilities of the outcomes from two dice. The probabilities are as follows.

• Probability of getting a doublet(Same number) = 6/36 = 1/6
• Probability of getting a number 3 on at least one dice = 11/36
• Probability of getting a sum of 7 = 6/36 = 1/6

As we see, when we roll a single die, there are 6 possibilities. When we roll two dice, there are 36 (= 6 2 ) possibilities. When we roll 3 dice we get 216 (= 6 3 ) possibilities. So a general formula to represent the number of outcomes on rolling 'n' dice is 6 n .

Probability of Drawing Cards

A deck containing 52 cards is grouped into four suits of clubs, diamonds, hearts, and spades. Each of the clubs, diamonds, hearts, and spades have 13 cards each, which sum up to 52. Now let us discuss the probability of drawing cards from a pack. The symbols on the cards are shown below. Spades and clubs are black cards. Hearts and diamonds are red cards.

The 13 cards in each suit are ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king. In these, the jack, the queen, and the king are called face cards. We can understand the card probability from the following examples.

• The probability of drawing a black card is P(Black card) = 26/52 = 1/2
• The probability of drawing a hearts card is P(Hearts) = 13/52 = 1/4
• The probability of drawing a face card is P(Face card) = 12/52 = 3/13
• The probability of drawing a card numbered 4 is P(4) = 4/52 = 1/13
• The probability of drawing a red card numbered 4 is P(4 Red) = 2/52 = 1/26

Probability Theorems

The following theorems of probability are helpful to understand the applications of probability and also perform the numerous calculations involving probability.

Theorem 1: The sum of the probability of happening of an event and not happening of an event is equal to 1. P(A) + P(A') = 1.

Theorem 2: The probability of an impossible event or the probability of an event not happening is always equal to 0. P(ϕ) = 0.

Theorem 3: The probability of a sure event is always equal to 1. P(A) = 1

Theorem 4: The probability of happening of any event always lies between 0 and 1. 0 < P(A) < 1

Theorem 5: If there are two events A and B, we can apply the formula of the union of two sets and we can derive the formula for the probability of happening of event A or event B as follows.

P(A∪B) = P(A) + P(B) - P(A∩B)

Also for two mutually exclusive events A and B, we have P( A U B) = P(A) + P(B)

Bayes' Theorem on Conditional Probability

Bayes' theorem describes the probability of an event based on the condition of occurrence of other events. It is also called conditional probability . It helps in calculating the probability of happening of one event based on the condition of happening of another event.

For example, let us assume that there are three bags with each bag containing some blue, green, and yellow balls. What is the probability of picking a yellow ball from the third bag? Since there are blue and green colored balls also, we can arrive at the probability based on these conditions also. Such a probability is called conditional probability.

The formula for Bayes' theorem is \(\begin{align}P(A|B) = \dfrac{ P(B|A)·P(A)} {P(B)}\end{align}\)

where, \(\begin{align}P(A|B) \end{align}\) denotes how often event A happens on a condition that B happens.

where, \(\begin{align}P(B|A) \end{align}\) denotes how often event B happens on a condition that A happens.

\(\begin{align}P(A) \end{align}\) the likelihood of occurrence of event A.

\(\begin{align}P(B) \end{align}\) the likelihood of occurrence of event B.

Law of Total Probability

If there are n number of events in an experiment, then the sum of the probabilities of those n events is always equal to 1.

P(A 1 ) + P(A 2 ) + P(A 3 ) + … + P(A n ) = 1

Important Notes on Probability:

• Probability is a measure of how likely an event is to happen.
• Probability is represented as a fraction and always lies between 0 and 1.
• An event can be defined as a subset of sample space.
• The sample of throwing a coin is {head, tail} and the sample space of throwing dice is {1, 2, 3, 4, 5, 6}.
• A random experiment cannot predict the exact outcomes but only some probable outcomes.

☛ Related Articles:

• Event Probability Calculator
• Probability and Statistics
• Probability Calculator

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Probability Examples

Example 1: What is the probability of getting a sum of 10 when two dice are thrown?

There are 36 possibilities when we throw two dice.

The desired outcome is 10. To get 10, we can have three favorable outcomes.

{(4,6),(6,4),(5,5)}

Probability of an event = number of favorable outcomes/ sample space

Probability of getting number 10 = 3/36 =1/12

Answer: Therefore the probability of getting a sum of 10 is 1/12.

Example 2: In a bag, there are 6 blue balls and 8 yellow balls. One ball is selected randomly from the bag. Find the probability of getting a blue ball.

Let us assume the probability of drawing a blue ball to be P(B)

Number of favorable outcomes to get a blue ball = 6

Total number of balls in the bag = 14

P(B) = Number of favorable outcomes/Total number of outcomes = 6/14 = 3/7

Answer: Therefore the probability of drawing a blue ball is 3/7.

Example 3: There are 5 cards numbered: 2, 3, 4, 5, 6. Find the probability of picking a prime number, and putting it back, you pick a composite number.

The two events are independent. Thus we use the product of the probability of the events.

P(getting a prime) = n(favorable events)/ n(sample space) = {2, 3, 5}/{2, 3, 4, 5, 6} = 3/5

p(getting a composite) = n(favorable events)/ n(sample space) = {4, 6}/{2, 3, 4, 5, 6}= 2/5

Thus the total probability of the two independent events = P(prime) × P(composite)

= 3/5 × (2/5)

Answer: Therefore the probability of picking a prime number and a prime number again is 6/25.

Example 4: Find the probability of getting a face card from a standard deck of cards using the probability equation.

Solution: To find: Probability of getting a face card Given: Total number of cards = 52 Number of face cards = Favorable outcomes = 12 Using the probability formula, Probability = (Favorable Outcomes)÷(Total Favourable Outcomes) P(face card) = 12/52 m = 3/13

Answer: The probability of getting a face card is 3/13

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Practice Questions on Probability

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FAQs on Probability

What is the meaning of probability in statistics.

Probability is a branch of math which deals with finding out the likelihood of the occurrence of an event. Probability measures the chance of an event happening and is equal to the number of favorable events divided by the total number of events. The value of probability ranges between 0 and 1, where 0 denotes uncertainty and 1 denotes certainty.

How to Find Probability?

The probability can be found by first knowing the sample space of the outcomes of an experiment. A probability is generally calculated for an event (x) within the sample space. The probability of an event happening is obtained by dividing the number of outcomes of an event by the total number of possible outcomes or sample space.

What are the Three Types of Probability?

The three types of probabilities are theoretical probability, experimental probability, and axiomatic probability. The theoretical probability calculates the probability based on formulas and input values. The experimental probability gives a realistic value and is based on the experimental values for calculation. Quite often the theoretical and experimental probability differ in their results. And the axiomatic probability is based on the axioms which govern the concepts of probability.

How To Calculate Probability?

The probability of any event depends upon the number of favorable outcomes and the total outcomes. Finding probability is finding the ratio of the number of favorable outcomes to the total outcomes in that sample space. It is expressed as, Probability of an event P(E) = (Number of favorable outcomes) ÷ (Number of Elements in Sample space).

What is Conditional Probability?

The conditional probability predicts the happening of one event based on the happening of another event. If there are two events A and B, conditional probability is a chance of occurrence of event B provided the event A has already occurred. The formula for the conditional probability of happening of event B, given that event A, has happened is P(B/A) = P(A ∩ B)/P(A).

What is Experimental Probability?

The experimental probability is based on the results and the values obtained from the probability experiments. Experimental probability is defined as the ratio of the total number of times an event has occurred to the total number of trials conducted. The results of the experimental probability are based on real-life instances and may differ in values from theoretical probability.

What is a Probability Distribution?

The two important probability distributions are binomial distribution and Poisson distribution. The binomial distribution is defined for events with two probability outcomes and for events with a multiple number of times of such events. The Poisson distribution is based on the numerous probability outcomes in a limited space of time, distance, sample space. An example of the binomial distribution is the tossing of a coin with two outcomes, and for conducting such a tossing experiment with n number of coins. A Poisson distribution is for events such as antigen detection in a plasma sample, where the probabilities are numerous.

How are Probability and Statistics Related?

The probability calculates the happening of an experiment and it calculates the happening of a particular event with respect to the entire set of events. For simple events of a few numbers of events, it is easy to calculate the probability. But for calculating probabilities involving numerous events and to manage huge data relating to those events we need the help of statistics . Statistics helps in rightly analyzing

How Probability is Used in Real Life?

Probability has huge applications in games and analysis. Also in real life and industry areas where it is about prediction we make use of probability. The prediction of the price of a stock, or the performance of a team in cricket requires the use of probability concepts. Further, the new technology field of artificial intelligence is extensively based on probability.

Where Do We Use the Probability Formula In Our Real Life?

The following activities in our real-life tend to follow the probability equation:

• Weather forecasting
• Playing cards
• Voting strategy in politics
• Rolling a dice.
• Pulling out the exact matching socks of the same color
• Chances of winning or losing in any sports.

How was Probability Discovered?

The use of the word "probable" started first in the seventeenth century when it was referred to actions or opinions which were held by sensible people. Further, the word probable in the legal content was referred to a proposition that had tangible proof. The field of permutations and combinations, statistical inference, cryptoanalysis, frequency analysis have altogether contributed to this current field of probability.

What is the Conditional Probability Formula?

The conditional probability depends upon the happening of one event based on the happening of another event. The conditional probability formula of happening of event B, given that event A, has already happened is expressed as P(B/A) = P(A ∩ B)/P(A).

Teach yourself statistics

What is a Statistical Experiment?

All statistical experiments have three things in common:

• The experiment can have more than one possible outcome.
• Each possible outcome can be specified in advance.
• The outcome of the experiment depends on chance.

A coin toss has all the attributes of a statistical experiment. There is more than one possible outcome. We can specify each possible outcome (i.e., heads or tails) in advance. And there is an element of chance, since the outcome is uncertain.

The Sample Space

• A sample space is a set of elements that represents all possible outcomes of a statistical experiment.
• A sample point is an element of a sample space.
• An event is a subset of a sample space - one or more sample points.

Probability of an Event

With some statistical experiments, each sample point is equally likely to occur. In this situation, the probability of an event is very easy to compute. It is:

Think about the toss of a single die. The sample space consists of six possible outcomes (1, 2, 3, 4, 5, and 6). And each outcome is equally likely to occur. Suppose we defined Event A to be the die landing on an odd number. There are three odd numbers (1, 3, and 5). So, the probability of Event A would be 3/6 or 0.5.

Types of events

• Two events are mutually exclusive if they have no sample points in common.
• Two events are independent when the occurrence of one does not affect the probability of the occurrence of the other.

Test Your Understanding

• Suppose I roll a die. Is that a statistical experiment? Yes. Like a coin toss, rolling dice is a statistical experiment. There is more than one possible outcome. We can specify each possible outcome in advance. And there is an element of chance.
• When you roll a single die, what is the sample space? The sample space is all of the possible outcomes - an integer between 1 and 6.
• Which of the following are sample points when you roll a die - 3, 6, and 9? The numbers 3 and 6 are sample points, because they are in the sample space. The number 9 is not a sample point, since it is outside the sample space; with one die, the largest number that you can roll is 6.
• Which of the following sets represent an event when you roll a die? A.   {1} B.   {2, 4,} C.   {2, 4, 6} D.   All of the above The correct answer is D. Remember that an event is a subset of a sample space. The sample space is any integer from 1 to 6. Each of the sets shown above is a subset of the sample space, so each represents an event.
• Consider the events listed below. Which are mutually exclusive? A.   {1} B.   {2, 4,} C.   {2, 4, 6} Two events are mutually exclusive, if they have no sample points in common. Events A and B are mutually exclusive, and Events A and C are mutually exclusive; since they have no points in common. Events B and C have common sample points, so they are not mutually exclusive.
• Suppose you roll a die two times. Is each roll of the die an independent event? Yes. Two events are independent when the occurrence of one has no effect on the probability of the occurrence of the other. Neither roll of the die affects the outcome of the other roll; so each roll of the die is independent.

Module 3: Probability

The terminology of probability, learning outcomes.

• Understand and use the terminology of probability

Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity. An  experiment is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a chance experiment . Flipping one fair coin twice is an example of an experiment.

A result of an experiment is called an  outcome . The sample space of an experiment is the set of all possible outcomes. Three ways to represent a sample space are: to list the possible outcomes, to create a tree diagram, or to create a Venn diagram. The uppercase letter [latex]S[/latex] is used to denote the sample space. For example, if you flip one fair coin, [latex]S[/latex] = {[latex]H[/latex], [latex]T[/latex]} where [latex]H[/latex] = heads and [latex]T[/latex] = tails are the outcomes.

An  event is any combination of outcomes. Upper case letters like [latex]A[/latex] and [latex]B[/latex] represent events. For example, if the experiment is to flip one fair coin, event [latex]A[/latex] might be getting at most one head. The probability of an event [latex]A[/latex] is written [latex]P[/latex]([latex]A[/latex]).

The  probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, inclusive (that is, zero and one and all numbers between these values). [latex]P[/latex]([latex]A[/latex]) = [latex]0[/latex] means the event [latex]A[/latex] can never happen. [latex]P[/latex]([latex]A[/latex]) = [latex]1[/latex] means the event [latex]A[/latex] always happens. [latex]P[/latex]([latex]A[/latex]) = [latex]0.5[/latex] means the event [latex]A[/latex] is equally likely to occur or not to occur. For example, if you flip one fair coin repeatedly (from [latex]20[/latex] to [latex]2,000[/latex] to [latex]20,000[/latex] times) the relative frequency of heads approaches [latex]0.5[/latex] (the probability of heads).

Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair , six-sided die, each face ([latex]1, 2, 3, 4, 5, \text{or}\,6[/latex]) is as likely to occur as any other face. If you toss a fair coin, a Head ([latex]H[/latex]) and a Tail ([latex]T[/latex]) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

To calculate the probability of an event [latex]A[/latex] when all outcomes in the sample space are equally likely , count the number of outcomes for event [latex]A[/latex] and divide by the total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the sample space is {[latex]HH[/latex], [latex]TH[/latex], [latex]HT[/latex], [latex]TT[/latex]} where [latex]T[/latex] = tails and [latex]H[/latex] = heads. The sample space has four outcomes. [latex]A[/latex] = getting one head. There are two outcomes that meet this condition {[latex]HT[/latex], [latex]TH[/latex]}, so [latex]\displaystyle{P}{({A})}=\frac{{2}}{{4}}={0.5}[/latex].

Suppose you roll one fair six-sided die, with the numbers {[latex]1, 2, 3, 4, 5, 6[/latex]} on its faces. Let event [latex]E[/latex] = rolling a number that is at least five. There are two outcomes {[latex]5, 6[/latex]}. [latex]\displaystyle{P}{({E})}=\frac{{2}}{{6}}[/latex] as the number of repetitions grows larger and larger.

This important characteristic of probability experiments is known as the  law of large numbers which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word empirical is often used instead of the word observed.)

This video gives more examples of basic probabilities.

It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be  unfair , or biased . Two math professors in Europe had their statistics students test the Belgian one Euro coin and discovered that in [latex]250[/latex] trials, a head was obtained [latex]56[/latex]% of the time and a tail was obtained [latex]44[/latex]% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.

“OR” Event

An outcome is in the event [latex]A[/latex] OR [latex]B[/latex] if the outcome is in [latex]A[/latex] or is in [latex]B[/latex] or is in both [latex]A[/latex] and [latex]B[/latex]. For example, let [latex]A[/latex] = {[latex]1, 2, 3, 4, 5[/latex]} and [latex]B[/latex] = {[latex]4, 5, 6, 7, 8[/latex]}. [latex]A[/latex] OR [latex]B[/latex] = {[latex]1, 2, 3, 4, 5, 6, 7, 8[/latex]}. Notice that [latex]4[/latex] and [latex]5[/latex] are NOT listed twice.

“AND” Event

An outcome is in the event [latex]A[/latex] AND [latex]B[/latex] if the outcome is in both [latex]A[/latex] and [latex]B[/latex] at the same time. For example, let [latex]A[/latex] and [latex]B[/latex] be {[latex]1, 2, 3, 4, 5[/latex]} and {[latex]4, 5, 6, 7, 8[/latex]}, respectively. Then [latex]A[/latex] AND [latex]B[/latex] = {[latex]4, 5[/latex]}.

The  complement of event [latex]A[/latex] is denoted [latex]A'[/latex] (read “[latex]A[/latex] prime”). [latex]A'[/latex] consists of all outcomes that are NOT in [latex]A[/latex]. Notice that [latex]P[/latex]([latex]A[/latex]) + [latex]P[/latex]([latex]A'[/latex]) = [latex]1[/latex]. For example, let [latex]S[/latex] = {[latex]1, 2, 3, 4, 5, 6[/latex]} and let [latex]A[/latex] = {[latex]1, 2, 3, 4[/latex]}. Then, [latex]A'={5, 6}[/latex]. [latex]P(A) = \frac{{4}}{{6}}[/latex] and [latex]P(A') = \frac{{2}}{{6}}[/latex], and [latex]P(A) +P(A') =\frac{{4}}{{6}}+\frac{{2}}{{6}}={1}[/latex].

The  conditional probability of [latex]A[/latex] given [latex]B[/latex] is written [latex]P[/latex]([latex]A[/latex]|[latex]B[/latex]). [latex]P[/latex]([latex]A[/latex]|[latex]B[/latex]) is the probability that event [latex]A[/latex] will occur given that the event [latex]B[/latex] has already occurred. A conditional reduces the sample space. We calculate the probability of [latex]A[/latex] from the reduced sample space [latex]B[/latex]. The formula to calculate [latex]P[/latex]([latex]A[/latex]|[latex]B[/latex]) is [latex]\displaystyle{P}{({A}{|}{B})}=\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({B})}}}[/latex] where [latex]P[/latex]([latex]B[/latex]) is greater than zero.

For example, suppose we toss one fair, six-sided die. The sample space [latex]S[/latex] = {[latex]1, 2, 3, 4, 5, 6[/latex]}. Let [latex]A[/latex] = face is [latex]2[/latex] or [latex]3[/latex] and [latex]B[/latex] = face is even ([latex]2, 4, 6[/latex]). To calculate [latex]P[/latex]([latex]A[/latex]|[latex]B[/latex]), we count the number of outcomes [latex]2[/latex] or [latex]3[/latex] in the sample space [latex]B[/latex] = {[latex]2, 4, 6[/latex]}. Then we divide that by the number of outcomes [latex]B[/latex] (rather than [latex]S[/latex]).

We get the same result by using the formula. Remember that [latex]S[/latex] has six outcomes.

[latex]\displaystyle{P}{({A}{|}{B})}=\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({B})}}}=\frac{{\frac{{\text{the number of outcomes that are 2 or 3 and even in } {S}}}{{6}}}}{{\frac{{\text{the number of outcomes that are even in } {S}}}{{6}}}}=\frac{{\frac{{1}}{{6}}}}{{\frac{{3}}{{6}}}}=\frac{{1}}{{3}}[/latex]

Understanding Terminology and Symbols

It is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.

The sample space [latex]S[/latex] is the whole numbers starting at one and less than [latex]20[/latex].

• [latex]S[/latex] = _____________________________Let event [latex]A[/latex] = the even numbers and event [latex]B[/latex] = numbers greater than [latex]13[/latex].
• [latex]A[/latex] = _____________________, [latex]B[/latex] = _____________________
• [latex]P[/latex]([latex]A[/latex]) = _____________, [latex]P[/latex]([latex]B[/latex]) = ________________
• [latex]A[/latex] AND [latex]B[/latex] = ____________________, [latex]A[/latex] OR [latex]B[/latex] = ________________
• [latex]P[/latex]([latex]A[/latex] AND [latex]B[/latex]) = _________, [latex]P[/latex]([latex]A[/latex] OR [latex]B[/latex]) = _____________
• [latex]A'[/latex] = _____________, [latex]P[/latex]([latex]A'[/latex]) = _____________
• [latex]P[/latex]([latex]A[/latex]) + [latex]P[/latex]([latex]A'[/latex]) = ____________
• [latex]P[/latex]([latex]A[/latex]|[latex]B[/latex]) = ___________, [latex]P[/latex]([latex]B[/latex]|[latex]A[/latex]) = _____________; are the probabilities equal?
• [latex]S[/latex] = {[latex]1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19[/latex]}
• [latex]A[/latex] = {[latex]2, 4, 6, 8, 10, 12, 14, 16, 18[/latex]}, [latex]B[/latex] = {[latex]14, 15, 16, 17, 18, 19[/latex]}
• [latex]\displaystyle{P}{({A})}=\frac{{9}}{{19}},{P}{({B})}=\frac{{6}}{{19}}[/latex]
• [latex]A[/latex] AND [latex]B[/latex] = {[latex]14,16,18[/latex]}, [latex]A[/latex] OR [latex]B[/latex] = [latex]2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19[/latex]}
• [latex]\displaystyle{P}{({A}\text{ AND } {B})}=\frac{{3}}{{19}},{P}{({A}\text{ OR } {B})}=\frac{{12}}{{19}}[/latex]
• [latex]\displaystyle{A'}={1},{3},{5},{7},{9},{11},{13},{15},{17},{19};{P}{({A}′)}=\frac{{10}}{{19}}[/latex]
• [latex]P[/latex]([latex]A[/latex]) + [latex]P[/latex]([latex]A'[/latex])=[latex]\displaystyle{1}{\left(\frac{{9}}{{19}}+\frac{{10}}{{19}}\right)={1}}[/latex]
• [latex]P[/latex]([latex]A[/latex]|[latex]B[/latex]) = [latex]\displaystyle\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({B})}}}=\frac{{3}}{{6}}[/latex],[latex]P[/latex]([latex]B[/latex]|[latex]A[/latex])=[latex]\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({A})}}}=\frac{{3}}{{9}}[/latex], No

The sample space [latex]S[/latex] is the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: ([latex]1, 4[/latex])).

• [latex]S[/latex] = _____________________________Let event [latex]A[/latex] = the sum is even and event [latex]B[/latex] = the first number is prime.
• [latex]B'[/latex] = _____________, [latex]P[/latex]([latex]B'[/latex]) = _____________
• [latex]P[/latex]([latex]A[/latex]) + [latex]P[/latex]([latex]A[/latex]) = ____________
• [latex]S[/latex] = {[latex](1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)[/latex]}
• [latex]A[/latex] = {[latex](1,1), (1,3), (2,2), (2,4), (3,1), (3,3)[/latex]} [latex]B[/latex] = {[latex](2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)[/latex]}
• [latex]\displaystyle{P}{({A})}=\frac{{1}}{{2}},{P}{({B})}=\frac{{2}}{{3}}[/latex]
• [latex]A[/latex] AND [latex]B[/latex] = {[latex](2,2), (2,4), (3,1), (3,3)[/latex]} [latex]A[/latex] OR [latex]B[/latex] = {[latex](1,1), (1,3), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)[/latex]}
• [latex]P[/latex]([latex]A[/latex] AND [latex]B[/latex])=[latex]\displaystyle\frac{{1}}{{3}}[/latex], {[latex]P[/latex]([latex]A[/latex] OR [latex]B[/latex])=[latex]\frac{{5}}{{6}}[/latex]
• [latex]B'[/latex]={[latex](1,1),(1,2),(1,3),(1,4)[/latex]}, [latex]P[/latex]([latex]B'[/latex])=[latex]\frac{{1}}{{3}}[/latex]
• [latex]P[/latex]([latex]B[/latex]) + [latex]P[/latex]([latex]B'[/latex]) = [latex]1[/latex]
• [latex]P[/latex]([latex]A[/latex]|[latex]B[/latex])=[latex]\displaystyle=\frac{{2}}{{3}}[/latex], No.

A fair, six-sided die is rolled. Describe the sample space [latex]S[/latex], identify each of the following events with a subset of [latex]S[/latex] and compute its probability (an outcome is the number of dots that show up).

• Event [latex]T[/latex] = the outcome is two.
• Event [latex]A[/latex] = the outcome is an even number.
• Event [latex]B[/latex] = the outcome is less than four.
• The complement of [latex]A[/latex].
• [latex]A[/latex] GIVEN [latex]B[/latex]
• [latex]B[/latex] GIVEN [latex]A[/latex]
• [latex]A[/latex] AND [latex]B[/latex]
• [latex]A[/latex] OR [latex]B[/latex]
• [latex]A[/latex] OR [latex]B'[/latex]
• Event [latex]N[/latex] = the outcome is a prime number.
• Event [latex]I[/latex] = the outcome is seven.
• [latex]T[/latex] = {[latex]2[/latex]}, [latex]P[/latex]([latex]T[/latex]) =[latex]\displaystyle\frac{{1}}{{6}}[/latex]
• [latex]A[/latex] = {[latex]2, 4, 6[/latex]} , [latex]P[/latex]([latex]A[/latex])=[latex]\displaystyle\frac{{1}}{{2}}[/latex]
• [latex]B[/latex] = {[latex]1, 2, 3[/latex]},[latex]P[/latex]([latex]B[/latex])=[latex]\displaystyle\frac{{1}}{{2}}[/latex]
• [latex]A'[/latex] = {[latex]1, 3,5[/latex]}, [latex]P[/latex]([latex]A'[/latex])=[latex]\displaystyle\frac{{1}}{{2}}[/latex]
• ([latex]A[/latex] | [latex]B[/latex])={[latex]2[/latex]},[latex]P[/latex]([latex]A[/latex] | [latex]B[/latex]) =[latex]\displaystyle\frac{{1}}{{3}}[/latex]
•  ( [latex]B[/latex] | [latex]A[/latex])  ={[latex]2[/latex]},[latex]P[/latex]([latex]B[/latex]|[latex]A[/latex])=[latex]\displaystyle\frac{{1}}{{3}}[/latex]
• ([latex]A[/latex] and [latex]B[/latex])={[latex]2[/latex]}, [latex]P[/latex]([latex]A[/latex] and [latex]B[/latex])=[latex]\displaystyle\frac{{1}}{{6}}[/latex]
• ([latex]A[/latex] or[latex]B[/latex])={[latex]1, 2, 3, 4,6[/latex]},[latex]P[/latex]([latex]A[/latex] or [latex]B[/latex])=[latex]\displaystyle\frac{{5}}{{6}}[/latex]
• ([latex]A[/latex] or [latex]B'[/latex])={[latex]2, 4, 5,6[/latex]},[latex]P[/latex]([latex]A[/latex] or [latex]B'[/latex])=[latex]\displaystyle\frac{{2}}{{3}}[/latex]
• [latex]N[/latex] = {[latex]2, 3,5[/latex]}, [latex]P[/latex]([latex]N[/latex])=[latex]\displaystyle\frac{{1}}{{2}}[/latex]
• A six-sided die does not have seven dots. [latex]P[/latex]([latex]7[/latex]) = [latex]0[/latex].

The table describes the distribution of a random sample [latex]S[/latex] of [latex]100[/latex] individuals, organized by gender and whether they are right- or left-handed.

Let’s denote the events [latex]M[/latex] = the subject is male,[latex]F[/latex] = the subject is female, [latex]R[/latex] = the subject is right-handed, [latex]L[/latex] = the subject is left-handed. Compute the following probabilities:

• [latex]P[/latex]([latex]M[/latex])
• [latex]P[/latex]([latex]F[/latex])
• [latex]P[/latex]([latex]R[/latex])
• [latex]P[/latex]([latex]L[/latex])
• [latex]P[/latex]([latex]M[/latex] AND [latex]R[/latex])
• [latex]P[/latex]([latex]F[/latex] AND [latex]L[/latex])
• [latex]P[/latex]([latex]M[/latex] OR [latex]F[/latex])
• [latex]P[/latex]([latex]M[/latex] OR [latex]R[/latex])
• [latex]P[/latex]([latex]F[/latex] OR [latex]L[/latex])
• [latex]P[/latex]([latex]M'[/latex])
• [latex]P[/latex]([latex]R[/latex]|[latex]M[/latex])
• [latex]P[/latex]([latex]F[/latex]|[latex]L[/latex])
• [latex]P[/latex]([latex]L[/latex]|[latex]F[/latex])
• [latex]P[/latex]([latex]M[/latex]) = [latex]0.52[/latex]
• [latex]P[/latex]([latex]F[/latex]) = [latex]0.48[/latex]
• [latex]P[/latex]([latex]R[/latex]) =[latex] 0.87[/latex]
• [latex]P[/latex]([latex]L[/latex]) = [latex]0.13[/latex]
• [latex]P[/latex]([latex]M[/latex] AND [latex]R[/latex]) = [latex]0.43[/latex]
• [latex]P[/latex]([latex]F[/latex] AND [latex]L[/latex]) = [latex]0.04[/latex]
• [latex]P[/latex]([latex]M[/latex] OR [latex]F[/latex]) = [latex]1[/latex]
• [latex]P[/latex]([latex]M[/latex] OR [latex]R[/latex]) = [latex]0.96[/latex]
• [latex]P[/latex]([latex]F[/latex] OR [latex]L[/latex]) = [latex]0.57[/latex]
• [latex]P[/latex]([latex]M'[/latex]) = [latex]0.48[/latex]
• [latex]P[/latex]([latex]R[/latex]|[latex]M[/latex]) = [latex]0.8269[/latex] (rounded to four decimal places)
• [latex]P[/latex]([latex]F[/latex]|[latex]L[/latex]) = [latex]0.3077[/latex] (rounded to four decimal places)
• [latex]P[/latex]([latex]L[/latex]|[latex]F[/latex]) = [latex]0.0833[/latex]

“Countries List by Continent.” Worldatlas, 2013. Available online at http://www.worldatlas.com/cntycont.htm (accessed May 2, 2013).

Concept Review

In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.

Formula Review

[latex]A[/latex] and [latex]B[/latex] are events

[latex]P[/latex]([latex]S[/latex]) = [latex]1[/latex] where [latex]S[/latex] is the sample space 0 ≤ [latex]P[/latex]([latex]A[/latex]) ≤ [latex]1[/latex]

[latex]P[/latex]([latex]A[/latex]|[latex]B[/latex])=[latex]\displaystyle\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({B})}}}[/latex]

• OpenStax, Statistics, Terminology. License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
• Introduction to Probability. Authored by : Mathispower4u. Located at : https://youtu.be/YWt_u5l_jHs . License : All Rights Reserved . License Terms : Sgtandard YouTube License

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Statistics and probability

Course: statistics and probability   >   unit 7.

• Intro to theoretical probability

Probability: the basics

• Simple probability: yellow marble
• Simple probability: non-blue marble
• Simple probability
• Intuitive sense of probabilities
• Comparing probabilities
• The Monty Hall problem

• The probability of an event can only be between 0 and 1 and can also be written as a percentage.
• The probability of event A ‍   is often written as P ( A ) ‍   .
• If P ( A ) > P ( B ) ‍   , then event A ‍   has a higher chance of occurring than event B ‍   .
• If P ( A ) = P ( B ) ‍   , then events A ‍   and B ‍    are equally likely to occur.

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Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

Heads (H) or Tails (T)

• the probability of the coin landing H is ½
• the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6 .

The probability of any one of them is 1 6

In general:

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1 6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 4 5 = 0.8

Probability Line

We can show probability on a Probability Line :

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads .

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index .

Some words have special meaning in Probability:

Experiment : a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Outcome: A possible result.

Example: "6" is one of the outcomes of a throw of a die.

Trial: A single performance of an experiment.

Example: I conducted a coin toss experiment. After 4 trials I got these results:

Three trials had the outcome "Head", and one trial had the outcome "Tail"

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

"King" is not a sample point. There are 4 Kings, so that is 4 different sample points.

There are 6 different sample points in that sample space.

Event: one or more outcomes of an experiment

Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a "5"

An event can include more than one outcome:

• Choosing a "King" from a deck of cards (any of the 4 Kings)
• Rolling an "even number" (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... ... ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points :

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

 After 100 Trials , Alex has 19 "double" Events ... is that close to what you would expect?

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4.2: Experiments Having Equally Likely Outcomes

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Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair , six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (\(\text{H}\)) and a Tail (\(\text{T}\)) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

To calculate the probability of an event A when all outcomes in the sample space are equally likely , count the number of outcomes for event \(\text{A}\) and divide by the total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the sample space is \(\{\text{HH, TH, HT,TT}\}\) where \(\text{T} =\) tails and \(\text{H} =\) heads. The sample space has four outcomes. \(\text{A} =\) getting one head. There are two outcomes that meet this condition \(\text{\{HT, TH\}}\), so \(P(\text{A}) = \frac{2}{4} = 0.5\).

Suppose you roll one fair six-sided die, with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event \(\text{E} =\) rolling a number that is at least five. There are two outcomes {5, 6}. \(P(\text{E}) = \frac{2}{6}\). If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall, \(\frac{2}{6}\) of the rolls would result in an outcome of "at least five". You would not expect exactly \(\frac{2}{6}\). The long-term relative frequency of obtaining this result would approach the theoretical probability of \(\frac{2}{6}\) as the number of repetitions grows larger and larger.

Definition: Law of Large Numbers

This important characteristic of probability experiments is known as the law of large numbers which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word empirical is often used instead of the word observed.)

It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair , or biased . Two math professors in Europe had their statistics students test the Belgian one Euro coin and discovered that in 250 trials, a head was obtained 56% of the time and a tail was obtained 44% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.

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Random Experiments

We may perform various activities in our daily existence, sometimes repeating the same actions though we get the same result every time. Suppose, in mathematics, we can directly say that the sum of all interior angles of a given quadrilateral is 360 degrees, even if we don’t know the type of quadrilateral and the measure of each internal angle. Also, we might perform several experimental activities, where the result may or may not be the same even when they are repeated under the same conditions. For example, when we toss a coin, it may turn up a tail or a head, but we are unsure which results will be obtained. These types of experiments are called random experiments.

Random Experiment in Probability

An activity that produces a result or an outcome is called an experiment. It is an element of uncertainty as to which one of these occurs when we perform an activity or experiment. Usually, we may get a different number of outcomes from an experiment. However, when an experiment satisfies the following two conditions, it is called a random experiment.

(i) It has more than one possible outcome.

(ii) It is not possible to predict the outcome in advance.

Let’s have a look at the terms involved in random experiments which we use frequently in probability theory. Also, these terms are used to describe whether an experiment is random or not.

Learn more about sample space here.

What is a Random Experiment?

Based on the definition of random experiment we can identify whether the given experiment is random or not. Go through the examples to understand what is a random experiment and what is not a random experiment.

Is picking a card from a well-shuffled deck of cards a random experiment?

We know that a deck contains 52 cards, and each of these cards has an equal chance to be selected.

(i) The experiment can be repeated since we can shuffle the deck of cards every time before picking a card and there are 52 possible outcomes.

(ii) It is possible to pick any of the 52 cards, and hence the outcome is not predictable before.

Thus, the given activity satisfies the two conditions of being a random experiment.

Hence, this is a random experiment.

Consider the experiment of dividing 36 by 4 using a calculator. Check whether it is a random experiment or not.

(i) This activity can be repeated under identical conditions though it has only one possible result.

(ii) The outcome is always 9, which means we can predict the outcome each time we repeat the operation.

Hence, the given activity is not a random experiment.

Examples of Random Experiments

Below are the examples of random experiments and the corresponding sample space.

Number of possible outcomes = 8

Number of possible outcomes = 36

Number of possible outcomes = 100

Similarly, we can write several examples which can be treated as random experiments.

Playing Cards

Probability theory is the systematic consideration of outcomes of a random experiment. As defined above, some of the experiments include rolling a die, tossing coins, and so on. There is another experiment of playing cards. Here, a deck of cards is considered as the sample space. For example, picking a black card from a well-shuffled deck is also considered an event of the experiment, where shuffling cards is treater as the experiment of probability.

A deck contains 52 cards, 26 are black, and 16 are red.

However, these playing cards are classified into 4 suits, namely Spades, Hearts, Diamonds, and Clubs. Each of these four suits contains 13 cards.

We can also classify the playing cards into 3 categories as:

Aces:  A deck contains 4 Aces, of which 1 of every suit. 

Face cards:  Kings, Queens, and Jacks in all four suits, also known as court cards.

Number cards:  All cards from 2 to 10 in any suit are called the number cards. 

• Spades and Clubs are black cards, whereas Hearts and Diamonds are red.
• 13 cards of each suit = 1 Ace + 3 face cards + 9 number cards
• The probability of drawing any card will always lie between 0 and 1.
• The number of spades, hearts, diamonds, and clubs is the same in every pack of 52 playing cards.

An example problem on picking a card from a deck is given above.

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Binomial Distribution Calculator

Use this binomial probability calculator to easily calculate binomial cumulative distribution function and probability mass given the probability on a single trial, the number of trials and events. It can calculate the probability of success if the outcome is a binomial random variable, for example if flipping a coin. The calculator can also solve for the number of trials required.

Related calculators

• Using the Binomial Probability Calculator
• What is a Binomial Distribution?
• What is a Binomial Probability?
• Binomial Cumulative Distribution Function (CDF)

    Using the Binomial Probability Calculator

You can use this tool to solve either for the exact probability of observing exactly x events in n trials, or the cumulative probability of observing X ≤ x , or the cumulative probabilities of observing X < x or X ≥ x or X > x. Simply enter the probability of observing an event (outcome of interest, success) on a single trial (e.g. as 0.5 or 1/2, 1/6 and so on), the number of trials and the number of events you want the probability calculated for. As long as the procedure generating the event conforms to the random variable model under a Binomial distribution the calculator applies. In other words, X must be a random variable generated by a process which results in Binomially-distributed, Independent and Identically Distributed outcomes (BiIID). For example, you can compute the probability of observing exactly 5 heads from 10 coin tosses of a fair coin (24.61%) , of rolling more than 2 sixes in a series of 20 dice rolls (67.13%) and so on. See more examples below.

Under the same conditions you can use the binomial probability distribution calculator above to compute the number of attempts you would need to see x or more outcomes of interest (successes, events). For example, if you know you have a 1% chance (1 in 100) to get a prize on each draw of a lottery, you can compute how many draws you need to participate in to be 99.99% certain you win at least 1 prize ( 917 draws ). Note that this example doesn't apply if you are buying tickets for a single lottery draw (the events are not independent).

Sequences of Bernoulli trials: trials in which the outcome is either 1 or 0 with the same probability on each trial result in and are modelled as binomial distribution so any such problem is one which can be solved using the above tool: it essentially doubles as a coin flip calculator .

    What is a Binomial Distribution?

The binomial distribution X~Bin(n,p) is a probability distribution which results from the number of events in a sequence of n independent experiments with a binary / Boolean outcome: true or false, yes or no, event or no event, success or failure. These are also known as Bernoulli trials and thus a Binomial distribution is the result of a sequence of Bernoulli trials. The parameters which describe it are n - number of independent experiments and p the probability of an event of interest in a single experiment. It is often used as a teaching device and the practical applications of probability theory and statistics due its many desirable properties such as a known standard deviation and easy to compute cumulative distribution function and inverse function.

The binomial probability distribution can be used to model the number of events in a sample of size n drawn with replacement from a population of size N, e.g. coin tosses, dice rolls, and so on. If the sampling is carried out without replacement they are no longer independent and the result is a hypergeometric distribution, although the binomial remains a decent approximation if N >> n.

The above is a randomly generated binomial distribution from 10,000 simulated binomial experiments, each with 10 Bernoulli trials with probability of observing an event of 0.2 (20%).

    What is a Binomial Probability?

A probability for a certain outcome from a binomial distribution is what is usually referred to as a "binomial probability". It can be calculated using the formula for the binomial probability distribution function (PDF), a.k.a. probability mass function (PMF): f(x) , as follows:

Note that the above equation is for the probability of observing exactly the specified outcome. However, often when searching for a binomial probability formula calculator people are actually looking to calculate the cumulative probability of a binomially-distributed random variable: the probability of observing x or less than x events (successes, outcomes of interest).

    Binomial Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of the Binomial distribution is what is needed when you need to compute the probability of observing less than or more than a certain number of events/outcomes/successes from a number of trials. The Binomial CDF formula is simple:

Therefore, the cumulative binomial probability is simply the sum of the probabilities for all events from 0 to x. Our binomial distribution calculator uses the formula above to calculate the cumulative probability of events less than or equal to x, less than x, greater than or equal to x and greater than x for you. These are all cumulative binomial probabilities.

The inverse function is required when computing the number of trials required to observe a certain number of events, or more, with a certain probability. For this we use the inverse normal distribution function which provides a good enough approximation.

    Examples

Example 1: Coin flipping. If a fair coin (p = 1/2 = 0.5) is tossed 100 times, what is the probability of observing exactly 50 heads? What is the probability of observing more than 50 heads? Entering 0.5 or 1/2 in the calculator and 100 for the number of trials and 50 for "Number of events" we get that the chance of seeing exactly 50 heads is just under 8% while the probability of observing more than 50 is a whopping 46%. While in an infinite number of coin flips a fair coin will tend to come up heads exactly 50% of the time, in any small number of flips it is highly unlikely to observe exactly 50% heads.

Example 2: Dice rolling. If a fair dice is thrown 10 times, what is the probability of throwing at least one six? We know that a dice has six sides so the probability of success in a single throw is 1/6. Thus, using n=10 and x=1 we can compute using the Binomial CDF that the chance of throwing at least one six (X ≥ 1) is 0.8385 or 83.85 percent.

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Binomial Distribution Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/binomial-probability-calculator.php URL [Accessed Date: 19 Jun, 2024].

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