probability homework 4

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Probability Worksheets

Navigate through this assortment of printable probability worksheets that includes exercises on basic probability based on more likely, less likely, equally likely, certain and impossible events, pdf worksheets based on identifying suitable events, simple spinner problems, for students in grade 4, grade 5, and grade 6. With the required introduction, the beginners get to further their knowledge with skills like probability on single coin, two coins, days in a week, months in a year, fair die, pair of dice, deck of cards, numbers and more. Mutually exclusive and inclusive events, probability on odds and other challenging probability worksheets are useful for grade 7, grade 8, and high school. Grab some of these probability worksheets for free!

Probability on Coins

Simple probability worksheets based on tossing single coin or two coins. Identify the proper sample space before finding probability.

Probability in a single coin toss

Probability in pair of coin - 1

Probability in pair of coin - 2

Probability on Days and Months

Fun filled worksheet pdfs based on days in a week and months in a year. Sample space is easy to find but care is required in identifying like events.

Days of a week

Months of a year - 1

Months of a year - 2

Probability on Fair Die

Fair die is numbered from 1 to 6. Understand the multiples, divisors and factors and apply it on these probability worksheets.

Simple numbers

Multiples and divisors

Mutually exclusive and inclusive

Probability on Pair of Dice

Sample space is little large which contains 36 elements. Write all of them in papers before start answering on probability questions for grade 7 and grade 8.

Based on numbers

Based on sum and difference

Based on multiples and divisors

Based on factors

Probability on Numbers

Students should learn the concepts of multiples, divisors and factors before start practicing these printable worksheets.

Probability on numbers - 1

Probability on numbers - 2

Probability on numbers - 3

Probability on numbers - 4

Probability on numbers - 5

Probability on Deck of Cards

Deck of cards contain 52 cards, 26 are black, 26 are red, four different flowers, each flower contain 13 cards such as A, 1, 2, ..., 10, J, Q, K.

Deck of cards worksheet - 1

Deck of cards worksheet - 2

Deck of cards worksheet - 3

Probability on Spinners

Interactive worksheets for 4th grade and 5th grade kids to understand the probability using spinners. Colorful spinners are included for more fun.

Spinner worksheets on numbers

Spinner worksheets on colors

Probability on Odds

Probability on odds worksheets can be broadly classifieds as favorable to the events or against the events.

Odds worksheet - 1

Odds worksheet - 2

Odds worksheet - 3

Probability on Independent and Dependent

Here comes our challenging probability worksheets set for 8th grade and high school students based on dependent and independent events with various real-life applications.

Based on deck of cards

Based on marbles

Based on cards

Probability on Different Events

Basic probability worksheets for beginners in 6th grade and 7th grade to understand the different type of events such as more likely, less likely, equally likely and so on.

Balls in container

Identify suitable events

Mutually inclusive and exclusive events

Related Worksheets

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» Permutation and Combination

» Venn Diagram

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Statistics 110: Probability

Strategic Practice and Homework Problems

Actively solving practice problems is essential for learning probability. Strategic practice problems are organized by concept, to test and reinforce understanding of that concept.  Homework problems  usually do not say which concepts are involved, and often require combining several concepts. Each of the Strategic Practice documents here contains a set of strategic practice problems, solutions to those problems, a homework assignment, and solutions to the homework assignment. Also included here are the exercises from the  book that are marked with an s, and solutions to those exercises. 

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Unit 1: Place value

Unit 2: addition, subtraction, and estimation, unit 3: multiply by 1-digit numbers, unit 4: multiply by 2-digit numbers, unit 5: division, unit 6: factors, multiples and patterns, unit 7: equivalent fractions and comparing fractions, unit 8: add and subtract fractions, unit 9: multiply fractions, unit 10: understand decimals, unit 11: plane figures, unit 12: measuring angles, unit 13: area and perimeter, unit 14: units of measurement.

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:

	Trial	Trial	Trial	Trial
Head	✔	✔		✔
Tail			✔

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:

Trial	Is it a Double?
{3,4}	No
{5,1}	No
{2,2}
{6,3}	No
...	...

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

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&

4.1 Probability Distribution Function (PDF) for a Discrete Random Variable

Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given in Table 4.30 .

	( )
3	0.05
4	0.40
5	0.30
6	0.15
7	0.10

• In words, define the random variable X .
• What does it mean that the values zero, one, and two are not included for x in the PDF?

4.2 Mean or Expected Value and Standard Deviation

A theater group holds a fund-raiser. It sells 100 raffle tickets for $5 apiece. Suppose you purchase four tickets. The prize is two passes to a Broadway show, worth a total of $150.

• What are you interested in here?
• List the values that X may take on.
• Construct a PDF.
• If this fund-raiser is repeated often and you always purchase four tickets, what would be your expected average winnings per raffle?

( )

A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails.

• If the card is a face card, and the coin lands on Heads, you win $6
• If the card is a face card, and the coin lands on Tails, you win $2
• If the card is not a face card, you lose $2, no matter what the coin shows.
• Find the expected value for this game (expected net gain or loss).
• Explain what your calculations indicate about your long-term average profits and losses on this game.
• Should you play this game to win money?

You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available to be sold in this lottery. In this lottery there are one $500 prize, two $100 prizes, and four $25 prizes. Find your expected gain or loss.

= $ net gain or loss ( )

Complete the PDF and answer the questions.

	( )	( )
0	0.3
1	0.2
2
3	0.4

• Find the probability that x = 2.
• Find the expected value.

Suppose that you are offered the following “deal.” You roll a die. If you roll a six, you win $10. If you roll a four or five, you win $5. If you roll a one, two, or three, you pay $6.

• What are you ultimately interested in here (the value of the roll or the money you win)?
• In words, define the Random Variable X .
• Over the long run of playing this game, what are your expected average winnings per game?
• Based on numerical values, should you take the deal? Explain your decision in complete sentences.

( )

A venture capitalist, willing to invest $1,000,000, has three investments to choose from. The first investment, a software company, has a 10% chance of returning $5,000,000 profit, a 30% chance of returning $1,000,000 profit, and a 60% chance of losing the million dollars. The second company, a hardware company, has a 20% chance of returning $3,000,000 profit, a 40% chance of returning $1,000,000 profit, and a 40% chance of losing the million dollars. The third company, a biotech firm, has a 10% chance of returning $6,000,000 profit, a 70% of no profit or loss, and a 20% chance of losing the million dollars.

• Construct a PDF for each investment.
• Find the expected value for each investment.
• Which is the safest investment? Why do you think so?
• Which is the riskiest investment? Why do you think so?
• Which investment has the highest expected return, on average?

Suppose that 20,000 married adults in the United States were randomly surveyed as to the number of children they have. The results are compiled and are used as theoretical probabilities. Let X = the number of children married people have.

	( )	( )
0	0.10
1	0.20
2	0.30
3
4	0.10
5	0.05
6 (or more)	0.05

• Find the probability that a married adult has three children.
• In words, what does the expected value in this example represent?
• Is it more likely that a married adult will have two to three children or four to six children? How do you know?

Suppose that the PDF for the number of years it takes to earn a Bachelor of Science (B.S.) degree is given as in Table 4.33 .

	( )
3	0.05
4	0.40
5	0.30
6	0.15
7	0.10

On average, how many years do you expect it to take for an individual to earn a B.S.?

People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given in the following table. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs.

	( )
0	0.03
1	0.50
2	0.24
3
4	0.07
5	0.04

• Describe the random variable X in words.
• Find the probability that a customer rents three DVDs.
• Find the probability that a customer rents at least four DVDs.

	( )
0	0.35
1	0.25
2	0.20
3	0.10
4	0.05
5	0.05

• At which store is the expected number of DVDs rented per customer higher?
• If Video to Go estimates that they will have 300 customers next week, how many DVDs do they expect to rent next week? Answer in sentence form.
• If Video to Go expects 300 customers next week, and Entertainment HQ projects that they will have 420 customers, for which store is the expected number of DVD rentals for next week higher? Explain.
• Which of the two video stores experiences more variation in the number of DVD rentals per customer? How do you know that?

A “friend” offers you the following “deal.” For a $10 fee, you may pick an envelope from a box containing 100 seemingly identical envelopes. However, each envelope contains a coupon for a free gift.

• Ten of the coupons are for a free gift worth $6.
• Eighty of the coupons are for a free gift worth $8.
• Six of the coupons are for a free gift worth $12.
• Four of the coupons are for a free gift worth $40.

Based upon the financial gain or loss over the long run, should you play the game?

• Yes, I expect to come out ahead in money.
• No, I expect to come out behind in money.
• It doesn’t matter. I expect to break even.

Florida State University has 14 statistics classes scheduled for its Summer 2013 term. One class has space available for 30 students, eight classes have space for 60 students, one class has space for 70 students, and four classes have space for 100 students.

• What is the average class size assuming each class is filled to capacity?
• Space is available for 980 students. Suppose that each class is filled to capacity and select a statistics student at random. Let the random variable X equal the size of the student’s class. Define the PDF for X .
• Find the mean of X .
• Find the standard deviation of X .

( ) – ) ( )

In a lottery, there are 250 prizes of $5, 50 prizes of $25, and ten prizes of $100. Assuming that 10,000 tickets are to be issued and sold, what is a fair price to charge to break even?

4.3 Binomial Distribution

According to a recent article the average number of babies born with significant hearing loss (deafness) is approximately two per 1,000 babies in a healthy baby nursery. The number climbs to an average of 30 per 1,000 babies in an intensive care nursery.

Suppose that 1,000 babies from healthy baby nurseries were randomly surveyed. Find the probability that exactly two babies were born deaf.

Use the following information to answer the next four exercises. Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu.

Define the random variable and list its possible values.

State the distribution of X .

Find the probability that at least four of the 25 patients actually have the flu.

On average, for every 25 patients calling in, how many do you expect to have the flu?

People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given Table 4.36 . There is five-video limit per customer at this store, so nobody ever rents more than five DVDs.

	( )
0	0.03
1	0.50
2	0.24
3
4	0.07
5	0.04

• Find the probability that a customer rents at most two DVDs.

A school newspaper reporter decides to randomly survey 12 students to see if they will attend Tet (Vietnamese New Year) festivities this year. Based on past years, she knows that 18% of students attend Tet festivities. We are interested in the number of students who will attend the festivities.

• Give the distribution of X . X ~ _____(_____,_____)
• How many of the 12 students do we expect to attend the festivities?
• Find the probability that at most four students will attend.
• Find the probability that more than two students will attend.

Use the following information to answer the next three exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games.

The expected number of wins for that upcoming month is:

• 382 1043 382 1043

Let X = the number of games won in that upcoming month.

What is the probability that the San Jose Sharks win six games in that upcoming month?

What is the probability that the San Jose Sharks win at least five games in that upcoming month

A student takes a ten-question true-false quiz, but did not study and randomly guesses each answer. Find the probability that the student passes the quiz with a grade of at least 70% of the questions correct.

A student takes a 32-question multiple-choice exam, but did not study and randomly guesses each answer. Each question has three possible choices for the answer. Find the probability that the student guesses more than 75% of the questions correctly.

Six different colored dice are rolled. Of interest is the number of dice that show a one.

• On average, how many dice would you expect to show a one?
• Find the probability that all six dice show a one.
• Is it more likely that three or that four dice will show a one? Use numbers to justify your answer numerically.

More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance learning courses.

• On average, how many schools would you expect to offer such courses?
• Find the probability that at most ten offer such courses.
• Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence.

Suppose that about 85% of graduating students attend their graduation. A group of 22 graduating students is randomly chosen.

• How many are expected to attend their graduation?
• Find the probability that 17 or 18 attend.
• Based on numerical values, would you be surprised if all 22 attended graduation? Justify your answer numerically.

At The Fencing Center, 60% of the fencers use the foil as their main weapon. We randomly survey 25 fencers at The Fencing Center. We are interested in the number of fencers who do not use the foil as their main weapon.

• How many are expected to not to use the foil as their main weapon?
• Find the probability that six do not use the foil as their main weapon.
• Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically.

Approximately 8% of students at a local high school participate in after-school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number who participated in after-school sports all four years of high school.

• How many seniors are expected to have participated in after-school sports all four years of high school?
• Based on numerical values, would you be surprised if none of the seniors participated in after-school sports all four years of high school? Justify your answer numerically.
• Based upon numerical values, is it more likely that four or that five of the seniors participated in after-school sports all four years of high school? Justify your answer numerically.

The chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. We are interested in the expected number of audits a person with that income has in a 20-year period. Assume each year is independent.

• How many audits are expected in a 20-year period?
• Find the probability that a person is not audited at all.
• Find the probability that a person is audited more than twice.

It has been estimated that only about 30% of California residents have adequate earthquake supplies. Suppose you randomly survey 11 California residents. We are interested in the number who have adequate earthquake supplies.

• What is the probability that at least eight have adequate earthquake supplies?
• Is it more likely that none or that all of the residents surveyed will have adequate earthquake supplies? Why?
• How many residents do you expect will have adequate earthquake supplies?

There are two similar games played for Chinese New Year and Vietnamese New Year. In the Chinese version, fair dice with numbers 1, 2, 3, 4, 5, and 6 are used, along with a board with those numbers. In the Vietnamese version, fair dice with pictures of a gourd, fish, rooster, crab, crayfish, and deer are used. The board has those six objects on it, also. We will play with bets being $1. The player places a bet on a number or object. The “house” rolls three dice. If none of the dice show the number or object that was bet, the house keeps the $1 bet. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his or her $1 bet, plus $1 profit. If two of the dice show the number or object bet (and the third die does not show it), the player gets back his or her $1 bet, plus $2 profit. If all three dice show the number or object bet, the player gets back his or her $1 bet, plus $3 profit. Let X = number of matches and Y = profit per game.

• List the values that Y may take on. Then, construct one PDF table that includes both X and Y and their probabilities.
• Calculate the average expected matches over the long run of playing this game for the player.
• Calculate the average expected earnings over the long run of playing this game for the player.
• Determine who has the advantage, the player or the house.

According to The World Bank, only 9% of the population of Uganda had access to electricity as of 2009. Suppose we randomly sample 150 people in Uganda. Let X = the number of people who have access to electricity.

• What is the probability distribution for X ?
• Using the formulas, calculate the mean and standard deviation of X .
• Use your calculator to find the probability that 15 people in the sample have access to electricity.
• Find the probability that at most ten people in the sample have access to electricity.
• Find the probability that more than 25 people in the sample have access to electricity.

The literacy rate for a nation measures the proportion of people age 15 and over that can read and write. The literacy rate in Afghanistan is 28.1%. Suppose you choose 15 people in Afghanistan at random. Let X = the number of people who are literate.

• Sketch a graph of the probability distribution of X .
• Using the formulas, calculate the (i) mean and (ii) standard deviation of X .
• Find the probability that more than five people in the sample are literate. Is it is more likely that three people or four people are literate.

4.4 Geometric Distribution

A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call.

• On average, how many dealerships would we expect her to have to call until she finds one that has the car?
• Find the probability that she must call at most four dealerships.
• Find the probability that she must call three or four dealerships.

Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each person is considered independent. We are interested in the number of adults in America we must survey until we find one who will watch the Super Bowl.

• How many adults in America do you expect to survey until you find one who will watch the Super Bowl?
• Find the probability that you must ask seven people.
• Find the probability that you must ask three or four people.

It has been estimated that only about 30% of California residents have adequate earthquake supplies. Suppose we are interested in the number of California residents we must survey until we find a resident who does not have adequate earthquake supplies.

• What is the probability that we must survey just one or two residents until we find a California resident who does not have adequate earthquake supplies?
• What is the probability that we must survey at least three California residents until we find a California resident who does not have adequate earthquake supplies?
• How many California residents do you expect to need to survey until you find a California resident who does not have adequate earthquake supplies?
• How many California residents do you expect to need to survey until you find a California resident who does have adequate earthquake supplies?

In one of its Spring catalogs, L.L. Bean® advertised footwear on 29 of its 192 catalog pages. Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. Each page may be picked more than once.

• How many pages do you expect to advertise footwear on them?
• Is it probable that all twenty will advertise footwear on them? Why or why not?
• What is the probability that fewer than ten will advertise footwear on them?
• Reminder: A page may be picked more than once. We are interested in the number of pages that we must randomly survey until we find one that has footwear advertised on it. Define the random variable X and give its distribution.
• What is the probability that you only need to survey at most three pages in order to find one that advertises footwear on it?
• How many pages do you expect to need to survey in order to find one that advertises footwear?

Suppose that you are performing the probability experiment of rolling one fair six-sided die. Let F be the event of rolling a four or a five. You are interested in how many times you need to roll the die in order to obtain the first four or five as the outcome.

• p = probability of success (event F occurs)
• q = probability of failure (event F does not occur)
• Write the description of the random variable X .
• What are the values that X can take on?
• Find the values of p and q .
• Find the probability that the first occurrence of event F (rolling a four or five) is on the second trial.

Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random. What values does X take on?

The World Bank records the prevalence of HIV in countries around the world. According to their data, “Prevalence of HIV refers to the percentage of people ages 15 to 49 who are infected with HIV.” 1 In South Africa, the prevalence of HIV is 17.3%. Let X = the number of people you test until you find a person infected with HIV.

• Sketch a graph of the distribution of the discrete random variable X .
• What is the probability that you must test 30 people to find one with HIV?
• What is the probability that you must ask ten people?
• Find the (i) mean and (ii) standard deviation of the distribution of X .

According to a recent Pew Research poll, 75% of millenials (people born between 1981 and 1995) have a profile on a social networking site. Let X = the number of millenials you ask until you find a person without a profile on a social networking site.

• Describe the distribution of X .
• Find the (i) mean and (ii) standard deviation of X .
• What is the probability that you must ask ten people to find one person without a social networking site?
• What is the probability that you must ask 20 people to find one person without a social networking site?
• What is the probability that you must ask at most five people?

4.5 Hypergeometric Distribution

A group of Martial Arts students is planning on participating in an upcoming demonstration. Six are students of Tae Kwon Do; seven are students of Shotokan Karate. Suppose that eight students are randomly picked to be in the first demonstration. We are interested in the number of Shotokan Karate students in that first demonstration.

• How many Shotokan Karate students do we expect to be in that first demonstration?

In one of its Spring catalogs, L.L. Bean® advertised footwear on 29 of its 192 catalog pages. Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. Each page may be picked at most once.

• Calculate the standard deviation.

Suppose that a technology task force is being formed to study technology awareness among instructors. Assume that ten people will be randomly chosen to be on the committee from a group of 28 volunteers, 20 who are technically proficient and eight who are not. We are interested in the number on the committee who are not technically proficient.

• How many instructors do you expect on the committee who are not technically proficient?
• Find the probability that at least five on the committee are not technically proficient.
• Find the probability that at most three on the committee are not technically proficient.

Suppose that nine Massachusetts athletes are scheduled to appear at a charity benefit. The nine are randomly chosen from eight volunteers from the Boston Celtics and four volunteers from the New England Patriots. We are interested in the number of Patriots picked.

• Are you choosing the nine athletes with or without replacement?

A bridge hand is defined as 13 cards selected at random and without replacement from a deck of 52 cards. In a standard deck of cards, there are 13 cards from each suit: hearts, spades, clubs, and diamonds. What is the probability of being dealt a hand that does not contain a heart?

• What is the group of interest?
• How many are in the group of interest?
• How many are in the other group?
• Let X = _________. What values does X take on?
• The probability question is P (_______).
• Find the probability in question.

4.6 Poisson Distribution

The switchboard in a Minneapolis law office gets an average of 5.5 incoming phone calls during the noon hour on Mondays. Experience shows that the existing staff can handle up to six calls in an hour. Let X = the number of calls received at noon.

• Find the mean and standard deviation of X .
• What is the probability that the office receives at most six calls at noon on Monday?
• Find the probability that the law office receives six calls at noon. What does this mean to the law office staff who get, on average, 5.5 incoming phone calls at noon?
• What is the probability that the office receives more than eight calls at noon?

The maternity ward at Dr. Jose Fabella Memorial Hospital in Manila in the Philippines is one of the busiest in the world with an average of 60 births per day. Let X = the number of births in an hour.

• What is the probability that the maternity ward will deliver three babies in one hour?
• What is the probability that the maternity ward will deliver at most three babies in one hour?
• What is the probability that the maternity ward will deliver more than five babies in one hour?

A manufacturer of Christmas tree light bulbs knows that 3% of its bulbs are defective. Find the probability that a string of 100 lights contains at most four defective bulbs using both the binomial and Poisson distributions.

The average number of children a Japanese woman has in her lifetime is 1.37. Suppose that one Japanese woman is randomly chosen.

• Find the probability that she has no children.
• Find the probability that she has fewer children than the Japanese average.
• Find the probability that she has more children than the Japanese average.

The average number of children a Spanish woman has in her lifetime is 1.47. Suppose that one Spanish woman is randomly chosen.

• Find the probability that she has fewer children than the Spanish average.
• Find the probability that she has more children than the Spanish average .

Fertile, female cats produce an average of three litters per year. Suppose that one fertile, female cat is randomly chosen. In one year, find the probability she produces:

• Give the distribution of X . X ~ _______
• Find the probability that she has no litters in one year.
• Find the probability that she has at least two litters in one year.
• Find the probability that she has exactly three litters in one year.

The chance of having an extra fortune in a fortune cookie is about 3%. Given a bag of 144 fortune cookies, we are interested in the number of cookies with an extra fortune. Two distributions may be used to solve this problem, but only use one distribution to solve the problem.

• How many cookies do we expect to have an extra fortune?
• Find the probability that none of the cookies have an extra fortune.
• Find the probability that more than three have an extra fortune.
• As n increases, what happens involving the probabilities using the two distributions? Explain in complete sentences.

According to the South Carolina Department of Mental Health web site, for every 200 U.S. women, the average number who suffer from anorexia is one. Out of a randomly chosen group of 600 U.S. women determine the following.

• How many are expected to suffer from anorexia?
• Find the probability that no one suffers from anorexia.
• Find the probability that more than four suffer from anorexia.

The chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. Suppose that 100 people with tax returns over $25,000 are randomly picked. We are interested in the number of people audited in one year. Use a Poisson distribution to anwer the following questions.

• How many are expected to be audited?
• Find the probability that no one was audited.
• Find the probability that at least three were audited.

Approximately 8% of students at a local high school participate in after-school sports all four years of high school. A group of 60 seniors is randomly chosen. Of interest is the number that participated in after-school sports all four years of high school.

• Based on numerical values, is it more likely that four or that five of the seniors participated in after-school sports all four years of high school? Justify your answer numerically.

On average, Pierre, an amateur chef, drops three pieces of egg shell into every two cake batters he makes. Suppose that you buy one of his cakes.

• On average, how many pieces of egg shell do you expect to be in the cake?
• What is the probability that there will not be any pieces of egg shell in the cake?
• Let’s say that you buy one of Pierre’s cakes each week for six weeks. What is the probability that there will not be any egg shell in any of the cakes?
• Based upon the average given for Pierre, is it possible for there to be seven pieces of shell in the cake? Why?

Use the following information to answer the next two exercises: The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is ten. We are interested in the number of times her cats wake her up each week.

In words, the random variable X = _________________

• the number of times Mrs. Plum’s cats wake her up each week.
• the number of times Mrs. Plum’s cats wake her up each hour.
• the number of times Mrs. Plum’s cats wake her up each night.
• the number of times Mrs. Plum’s cats wake her up.

Find the probability that her cats will wake her up no more than five times next week.

• 1 ”Prevalence of HIV, total (% of populations ages 15-49),” The World Bank, 2013. Available online at http://data.worldbank.org/indicator/SH.DYN.AIDS.ZS?order=wbapi_data_value_2011+wbapi_data_value+wbapi_data_value-last&sort=desc (accessed May 15, 2013).

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• Book title: Introductory Statistics
• Publication date: Sep 19, 2013
• Location: Houston, Texas
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Probability Calculator

Probability of two events.

To find out the union, intersection, and other related probabilities of two independent events.

Probability of A:
Probability of B:

Probability Solver for Two Events

Please provide any 2 values below to calculate the rest probabilities of two independent events.

Probability of A:
Probability of B:
Probability of A NOT occuring:
Probability of B NOT occuring:
Probability of A and B both occuring:
Probability that A or B or both occur:
Probability that A or B occurs but NOT both:
Probability of neither A nor B occuring:

Probability of a Series of Independent Events

	Probability	Repeat Times
Event A
Event B

Probability of a Normal Distribution

Use the calculator below to find the area P shown in the normal distribution, as well as the confidence intervals for a range of confidence levels.

):		For negative infinite, use -inf
):		For positive infinite, use inf

Related Standard Deviation Calculator | Sample Size Calculator | Statistics Calculator

Probability is the measure of the likelihood of an event occurring. It is quantified as a number between 0 and 1, with 1 signifying certainty, and 0 signifying that the event cannot occur. It follows that the higher the probability of an event, the more certain it is that the event will occur. In its most general case, probability can be defined numerically as the number of desired outcomes divided by the total number of outcomes. This is further affected by whether the events being studied are independent, mutually exclusive, or conditional, among other things. The calculator provided computes the probability that an event A or B does not occur, the probability A and/or B occur when they are not mutually exclusive, the probability that both event A and B occur, and the probability that either event A or event B occurs, but not both.

Complement of A and B

Given a probability A , denoted by P(A) , it is simple to calculate the complement, or the probability that the event described by P(A) does not occur, P(A') . If, for example, P(A) = 0.65 represents the probability that Bob does not do his homework, his teacher Sally can predict the probability that Bob does his homework as follows:

P(A') = 1 - P(A) = 1 - 0.65 = 0.35

Given this scenario, there is, therefore, a 35% chance that Bob does his homework. Any P(B') would be calculated in the same manner, and it is worth noting that in the calculator above, can be independent; i.e. if P(A) = 0.65, P(B) does not necessarily have to equal 0.35 , and can equal 0.30 or some other number.

Intersection of A and B

The intersection of events A and B , written as P(A ∩ B) or P(A AND B) is the joint probability of at least two events, shown below in a Venn diagram. In the case where A and B are mutually exclusive events, P(A ∩ B) = 0 . Consider the probability of rolling a 4 and 6 on a single roll of a die; it is not possible. These events would therefore be considered mutually exclusive. Computing P(A ∩ B) is simple if the events are independent. In this case, the probabilities of events A and B are multiplied. To find the probability that two separate rolls of a die result in 6 each time:

The calculator provided considers the case where the probabilities are independent. Calculating the probability is slightly more involved when the events are dependent, and involves an understanding of conditional probability, or the probability of event A given that event B has occurred, P(A|B) . Take the example of a bag of 10 marbles, 7 of which are black, and 3 of which are blue. Calculate the probability of drawing a black marble if a blue marble has been withdrawn without replacement (the blue marble is removed from the bag, reducing the total number of marbles in the bag):

Probability of drawing a blue marble:

P(A) = 3/10

Probability of drawing a black marble:

P(B) = 7/10

Probability of drawing a black marble given that a blue marble was drawn:

P(B|A) = 7/9

As can be seen, the probability that a black marble is drawn is affected by any previous event where a black or blue marble was drawn without replacement. Thus, if a person wanted to determine the probability of withdrawing a blue and then black marble from the bag:

Probability of drawing a blue and then black marble using the probabilities calculated above:

P(A ∩ B) = P(A) × P(B|A) = (3/10) × (7/9) = 0.2333

Union of A and B

In probability, the union of events, P(A U B) , essentially involves the condition where any or all of the events being considered occur, shown in the Venn diagram below. Note that P(A U B) can also be written as P(A OR B) . In this case, the "inclusive OR" is being used. This means that while at least one of the conditions within the union must hold true, all conditions can be simultaneously true. There are two cases for the union of events; the events are either mutually exclusive, or the events are not mutually exclusive. In the case where the events are mutually exclusive, the calculation of the probability is simpler:

A basic example of mutually exclusive events would be the rolling of a dice, where event A is the probability that an even number is rolled, and event B is the probability that an odd number is rolled. It is clear in this case that the events are mutually exclusive since a number cannot be both even and odd, so P(A U B) would be 3/6 + 3/6 = 1 , since a standard dice only has odd and even numbers.

The calculator above computes the other case, where the events A and B are not mutually exclusive. In this case:

Using the example of rolling dice again, find the probability that an even number or a number that is a multiple of 3 is rolled. Here the set is represented by the 6 values of the dice, written as:

	S = {1,2,3,4,5,6}
Probability of an even number:	P(A) = {2,4,6} = 3/6
Probability of a multiple of 3:	P(B) = {3,6} = 2/6
Intersection of A and B:	P(A ∩ B) = {6} = 1/6
	P(A U B) = 3/6 + 2/6 -1/6 = 2/3

Exclusive OR of A and B

Another possible scenario that the calculator above computes is P(A XOR B) , shown in the Venn diagram below. The "Exclusive OR" operation is defined as the event that A or B occurs, but not simultaneously. The equation is as follows:

As an example, imagine it is Halloween, and two buckets of candy are set outside the house, one containing Snickers, and the other containing Reese's. Multiple flashing neon signs are placed around the buckets of candy insisting that each trick-or-treater only takes one Snickers OR Reese's but not both! It is unlikely, however, that every child adheres to the flashing neon signs. Given a probability of Reese's being chosen as P(A) = 0.65 , or Snickers being chosen with P(B) = 0.349 , and a P(unlikely) = 0.001 that a child exercises restraint while considering the detriments of a potential future cavity, calculate the probability that Snickers or Reese's is chosen, but not both:

0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.999 - 0.4537 = 0.5453

Therefore, there is a 54.53% chance that Snickers or Reese's is chosen, but not both.

Normal Distribution

The normal distribution or Gaussian distribution is a continuous probability distribution that follows the function of:

where μ is the mean and σ 2 is the variance. Note that standard deviation is typically denoted as σ . Also, in the special case where μ = 0 and σ = 1 , the distribution is referred to as a standard normal distribution. Above, along with the calculator, is a diagram of a typical normal distribution curve.

The normal distribution is often used to describe and approximate any variable that tends to cluster around the mean, for example, the heights of male students in a college, the leaf sizes on a tree, the scores of a test, etc. Use the "Normal Distribution" calculator above to determine the probability of an event with a normal distribution lying between two given values (i.e. P in the diagram above); for example, the probability of the height of a male student is between 5 and 6 feet in a college. Finding P as shown in the above diagram involves standardizing the two desired values to a z-score by subtracting the given mean and dividing by the standard deviation, as well as using a Z-table to find probabilities for Z. If, for example, it is desired to find the probability that a student at a university has a height between 60 inches and 72 inches tall given a mean of 68 inches tall with a standard deviation of 4 inches, 60 and 72 inches would be standardized as such:

Given μ = 68; σ = 4 (60 - 68)/4 = -8/4 = -2 (72 - 68)/4 = 4/4 = 1

The graph above illustrates the area of interest in the normal distribution. In order to determine the probability represented by the shaded area of the graph, use the standard normal Z-table provided at the bottom of the page. Note that there are different types of standard normal Z-tables. The table below provides the probability that a statistic is between 0 and Z, where 0 is the mean in the standard normal distribution. There are also Z-tables that provide the probabilities left or right of Z, both of which can be used to calculate the desired probability by subtracting the relevant values.

For this example, to determine the probability of a value between 0 and 2, find 2 in the first column of the table, since this table by definition provides probabilities between the mean (which is 0 in the standard normal distribution) and the number of choices, in this case, 2. Note that since the value in question is 2.0, the table is read by lining up the 2 row with the 0 column, and reading the value therein. If, instead, the value in question were 2.11, the 2.1 row would be matched with the 0.01 column and the value would be 0.48257. Also, note that even though the actual value of interest is -2 on the graph, the table only provides positive values. Since the normal distribution is symmetrical, only the displacement is important, and a displacement of 0 to -2 or 0 to 2 is the same, and will have the same area under the curve. Thus, the probability of a value falling between 0 and 2 is 0.47725 , while a value between 0 and 1 has a probability of 0.34134. Since the desired area is between -2 and 1, the probabilities are added to yield 0.81859, or approximately 81.859%. Returning to the example, this means that there is an 81.859% chance in this case that a male student at the given university has a height between 60 and 72 inches.

The calculator also provides a table of confidence intervals for various confidence levels. Refer to the Sample Size Calculator for Proportions for a more detailed explanation of confidence intervals and levels. Briefly, a confidence interval is a way of estimating a population parameter that provides an interval of the parameter rather than a single value. A confidence interval is always qualified by a confidence level, usually expressed as a percentage such as 95%. It is an indicator of the reliability of the estimate.

z	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
	0	0.00399	0.00798	0.01197	0.01595	0.01994	0.02392	0.0279	0.03188	0.03586
	0.03983	0.0438	0.04776	0.05172	0.05567	0.05962	0.06356	0.06749	0.07142	0.07535
	0.07926	0.08317	0.08706	0.09095	0.09483	0.09871	0.10257	0.10642	0.11026	0.11409
	0.11791	0.12172	0.12552	0.1293	0.13307	0.13683	0.14058	0.14431	0.14803	0.15173
	0.15542	0.1591	0.16276	0.1664	0.17003	0.17364	0.17724	0.18082	0.18439	0.18793
	0.19146	0.19497	0.19847	0.20194	0.2054	0.20884	0.21226	0.21566	0.21904	0.2224
	0.22575	0.22907	0.23237	0.23565	0.23891	0.24215	0.24537	0.24857	0.25175	0.2549
	0.25804	0.26115	0.26424	0.2673	0.27035	0.27337	0.27637	0.27935	0.2823	0.28524
	0.28814	0.29103	0.29389	0.29673	0.29955	0.30234	0.30511	0.30785	0.31057	0.31327
	0.31594	0.31859	0.32121	0.32381	0.32639	0.32894	0.33147	0.33398	0.33646	0.33891
	0.34134	0.34375	0.34614	0.34849	0.35083	0.35314	0.35543	0.35769	0.35993	0.36214
	0.36433	0.3665	0.36864	0.37076	0.37286	0.37493	0.37698	0.379	0.381	0.38298
	0.38493	0.38686	0.38877	0.39065	0.39251	0.39435	0.39617	0.39796	0.39973	0.40147
	0.4032	0.4049	0.40658	0.40824	0.40988	0.41149	0.41308	0.41466	0.41621	0.41774
	0.41924	0.42073	0.4222	0.42364	0.42507	0.42647	0.42785	0.42922	0.43056	0.43189
	0.43319	0.43448	0.43574	0.43699	0.43822	0.43943	0.44062	0.44179	0.44295	0.44408
	0.4452	0.4463	0.44738	0.44845	0.4495	0.45053	0.45154	0.45254	0.45352	0.45449
	0.45543	0.45637	0.45728	0.45818	0.45907	0.45994	0.4608	0.46164	0.46246	0.46327
	0.46407	0.46485	0.46562	0.46638	0.46712	0.46784	0.46856	0.46926	0.46995	0.47062
	0.47128	0.47193	0.47257	0.4732	0.47381	0.47441	0.475	0.47558	0.47615	0.4767
	0.47725	0.47778	0.47831	0.47882	0.47932	0.47982	0.4803	0.48077	0.48124	0.48169
	0.48214	0.48257	0.483	0.48341	0.48382	0.48422	0.48461	0.485	0.48537	0.48574
	0.4861	0.48645	0.48679	0.48713	0.48745	0.48778	0.48809	0.4884	0.4887	0.48899
	0.48928	0.48956	0.48983	0.4901	0.49036	0.49061	0.49086	0.49111	0.49134	0.49158
	0.4918	0.49202	0.49224	0.49245	0.49266	0.49286	0.49305	0.49324	0.49343	0.49361
	0.49379	0.49396	0.49413	0.4943	0.49446	0.49461	0.49477	0.49492	0.49506	0.4952
	0.49534	0.49547	0.4956	0.49573	0.49585	0.49598	0.49609	0.49621	0.49632	0.49643
	0.49653	0.49664	0.49674	0.49683	0.49693	0.49702	0.49711	0.4972	0.49728	0.49736
	0.49744	0.49752	0.4976	0.49767	0.49774	0.49781	0.49788	0.49795	0.49801	0.49807
	0.49813	0.49819	0.49825	0.49831	0.49836	0.49841	0.49846	0.49851	0.49856	0.49861
	0.49865	0.49869	0.49874	0.49878	0.49882	0.49886	0.49889	0.49893	0.49896	0.499
	0.49903	0.49906	0.4991	0.49913	0.49916	0.49918	0.49921	0.49924	0.49926	0.49929
	0.49931	0.49934	0.49936	0.49938	0.4994	0.49942	0.49944	0.49946	0.49948	0.4995
	0.49952	0.49953	0.49955	0.49957	0.49958	0.4996	0.49961	0.49962	0.49964	0.49965
	0.49966	0.49968	0.49969	0.4997	0.49971	0.49972	0.49973	0.49974	0.49975	0.49976
	0.49977	0.49978	0.49978	0.49979	0.4998	0.49981	0.49981	0.49982	0.49983	0.49983
	0.49984	0.49985	0.49985	0.49986	0.49986	0.49987	0.49987	0.49988	0.49988	0.49989
	0.49989	0.4999	0.4999	0.4999	0.49991	0.49991	0.49992	0.49992	0.49992	0.49992
	0.49993	0.49993	0.49993	0.49994	0.49994	0.49994	0.49994	0.49995	0.49995	0.49995
	0.49995	0.49995	0.49996	0.49996	0.49996	0.49996	0.49996	0.49996	0.49997	0.49997
	0.49997	0.49997	0.49997	0.49997	0.49997	0.49997	0.49998	0.49998	0.49998	0.49998

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AI Homework Help: Math Solver 4+

Science and education app, joseph rodrick, designed for iphone.

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Presenting our app - Your Premier Homework Helper! This app is a comprehensive tool designed to streamline and enhance your academic experience. Explore its key features: - Snap and Solve: Simply snap a photo of your question and receive a detailed solution with step-by-step explanations. - Manual Input Assistance: Enter any question or problem manually and get a comprehensive answer with detailed breakdowns. - Essay Generation: Effortlessly create well-structured essays on specified topics. - Literary Summaries: Access concise summaries of any literary text. - Multi-Subject Problem Solving: Address questions in a broad range of subjects including Math, Chemistry, Physics, Biology, Economics, Finance, History, Geography, and more. - Grammar and Writing Checks: Achieve flawless writing with tools that check grammar and style. - Language Translations: Easily translate text into various languages. - AI Tutor Assistance: Consult our AI tutor for explanations on any topic or to clarify any aspect of the solutions provided. This app is your essential tool for academic success, equipped with a versatile set of features to boost your learning. Download the app and harness the power of technology at your fingertips! The app offers a subscription plan for unlimited access to all features. • Charges will apply to the iTunes Account at confirmation of purchase. • Subscriptions automatically renew unless auto-renew is turned off at least 24 hours before the end of the current period. • Accounts will be charged for renewal within 24 hours prior to the end of the current period, at the stated renewal cost. • Subscriptions and auto-renewal can be managed by going to the user's Account Settings after purchase. • Any unused portion of a free trial period, if offered, will be forfeited when the user subscribes to that publication, if applicable. Privacy Policy: https://docs.google.com/document/d/1W_fcdP0F1iEo_2MuM_UN1X0U7O8FOwVk7mKFatcff90/edit?usp=sharing Terms of Use: https://docs.google.com/document/d/1zHnuLVjg22sVZIeFsKqRlFSipDEegzD8O1k8kAuBeiA/edit?usp=sharing

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The developer, Joseph Rodrick , indicated that the app’s privacy practices may include handling of data as described below. For more information, see the developer’s privacy policy .

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A new issue of Annals of Probability has been published. You can get a summary in PDF-format at:

https://www.imstat.org/ publications/aop/aop_52_4/aop_ 52_4.pdf

If you or your library has subscribed to the journal, electronic access to the full journal articles is at: https://projecteuclid.org/  or  https://www.jstor.org/

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Power ranking MLB teams based on their World Series odds

As June comes to a close, so does the first half of the 2024 MLB season . The contenders have started to separate themselves from the pretenders in each league, and the playoff field is slowly taking shape.

However, the playoff pictures in each league have varied wildly in the way they've come into focus. While the divisional and wild card races in the American League are starting to shake out, there are only two teams in the National League that aren't within five games back of a wild card spot.

That chaotic energy in the National League has created an interesting buildup toward July's trade deadline and the second half of the season. BetMGM has tried to make sense of the chaos with their latest betting odds for the 2024 World Series winner.

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Here's how all 30 MLB teams rank by odds to win this year's World Series.

MLB power rankings: 2024 World Series odds

1. los angeles dodgers (+300).

When healthy, the top of the Dodgers' lineup features back-to-back-to-back MVPs: Mookie Betts, Shohei Ohtani and Freddie Freeman. This year, all three players rank in the top 11 of OPS. Ohtani leads MLB in batting average with his .322 mark and leads the National League with 25 home runs.

Almost every player outside of those three has been excellent as well. Catcher Will Smith is one of baseball's best offensive catchers, and outfielders Jason Heyward and Teoscar Hernández have experienced career resurgences since getting to Los Angeles.

And that's all with a pitching staff that ranks second in MLB in ERA (3.25) and WHIP (1.10) and first in batting average against (.214).

2. New York Yankees (+450)

Aaron Judge has matched his home run pace from his record-breaking 2022 season. Juan Soto has embraced superstardom in New York. Before his recent injury, Giancarlo Stanton looked the best he has in years. The high-powered offense as a whole has scored the third-most runs (408) in MLB this season and are second with 118 home runs.

At the same time, their pitching staff has stepped up to put together a strong season of its own. Marcus Stroman has been excellent, and outside of a couple of recent blow-up performances, so has rookie Luis Gil. The Yankees' staff ranks among the best in MLB this season and deserves as much praise as the team's offense.

3. Philadelphia Phillies (+500)

The Phillies came into this season off of back-to-back NLCS appearances, including their pennant-winning 2022 season. Somehow, this team looks even better than that of those last two years.

Philadelphia holds MLB's best record as of Thursday with a 53-27 mark. Their pitching staff leads MLB by far with its 3.07 ERA. The most impressive part? That mark isn't propped up by their traditionally outstanding pitchers – Zack Wheeler and Aaron Nola – but by Ranger Suárez (2.01 ERA, 2nd in MLB) and Cristopher Sánchez (2.67, 9th) and a bullpen that has been mostly lights-out.

That's all without mentioning the year Bryce Harper is having as he settles into his new position at first base. He's up to 20 home runs with a .987 OPS that ranks fifth in the big leagues. Oh, and his fielding range ranks in MLB's 93rd percentile with just two errors all year.

MLB: 9 key numbers from MLB's first half: Aaron Judge matching historic home run pace

4. Baltimore Orioles (+800)

There isn't a more exciting, young team in MLB than the Orioles.

Baltimore leads MLB in runs scored (417), home runs (132), slugging (.461), and OPS (.776). It also ranks third in pitching staff ERA (3.39), fourth in WHIP (1.16) and third in batting average against (.224).

The biggest standout on the Orioles this year has far and away been shortstop and leadoff hitter Gunnar Henderson, who turns 23 years old on Saturday and is second in MLB with 26 home runs. He trails only Judge and Ohtani with his 1.005 OPS this season and is a very real candidate to win AL MVP just one year after winning the league's Rookie of the Year Award.

5. Atlanta Braves (+900)

Losing reigning NL MVP Ronald Acuña for the rest of the season with a torn ACL was a massive blow to the Braves, but they are by no means out of the title race.

This is a team that has won 100 games in each of the last two seasons with the help of its talented core: second baseman Ozzie Albies, third baseman Austin Riley and first baseman Matt Olson. In addition, left fielder Jarred Kelenic seems to have turned the corner since joining the Braves this offseason. Marcell Ozuna has been the best designated hitter not named Shohei Ohtani in the league this year.

Atlanta's pitching has also been strong, with Reynaldo López leading MLB in ERA (1.70), and Chris Sale and Max Fried holding steady with ERAs below 3.10.

6. Seattle Mariners (+1400)

7. cleveland guardians (+1600), 8. houston astros (+2000), 9. minnesota twins (+2200), 10. milwaukee brewers (+2800), t-11. arizona diamondbacks, kansas city royals, san diego padres, texas rangers (+5000), t-15. boston red sox, st. louis cardinals (+6600), t-17. chicago cubs, new york mets, san francisco giants (+8000), 20. tampa bay rays (+10000), t-21. cincinnati reds, toronto blue jays (+12500), 23. pittsburgh pirates (+20000), 24. detroit tigers (+25000), t-25. los angeles angels, washington nationals (+50000), t-27. chicago white sox, colorado rockies, miami marlins, oakland athletics (+100000).

CLEVELAND GUARDIANS: Two years after All-Star career, Stephen Vogt managing AL Central leaders to AL's best record

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