Chapter 4: Discrete Random VariablesContent check — Statistics, Chapter 4: Discrete Random Variables

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, or in other words, how long new hires stay with the company. Over the years, the company has established the following probability distribution:

Let X = the number of years a new hire will stay with the company.

Let P(x) = the probability that a new hire will stay with the company x years.

Complete Table 4.20 using the data provided.

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, or in other words, how long new hires stay with the company. Over the years, the company has established the following probability distribution:

Let X = the number of years a new hire will stay with the company.

Let P(x) = the probability that a new hire will stay with the company x years.

P(x ≥ 5) = ________

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, or in other words, how long new hires stay with the company. Over the years, the company has established the following probability distribution:

Let X = the number of years a new hire will stay with the company.

Let P(x) = the probability that a new hire will stay with the company x years.

What does the column “P(x)” sum to?

Use the following information to answer the next four exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution.

What is the probability the baker will sell more than one batch? P(x > 1) = ________

Use the following information to answer the next four exercises: A baker is deciding how many batches of muffins to make to sell in his bakery. He wants to make enough to sell every one and no fewer. Through observation, the baker has established a probability distribution.

On average, how many batches should the baker make?

Use the following information to answer the next two exercises: Ellen has music practice three days a week. She practices for all of the three days 85 percent of the time, two days 8 percent of the time, one day 4 percent of the time, and no days 3 percent of the time. One week is selected at random.

Construct a probability distribution table for the data.

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35 percent of the time, four events 25 percent of the time, three events 20 percent of the time, two events 10 percent of the time, one event 5 percent of the time, and no events 5 percent of the time.

Define the random variable X.

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35 percent of the time, four events 25 percent of the time, three events 20 percent of the time, two events 10 percent of the time, one event 5 percent of the time, and no events 5 percent of the time.

Construct a PDF table.

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events 35 percent of the time, four events 25 percent of the time, three events 20 percent of the time, two events 10 percent of the time, one event 5 percent of the time, and no events 5 percent of the time.

Find the probability that Javier volunteers for at least one event each month. P(x > 0) = ________

Find the expected value from the expected value table.

Identify the mistake in the probability distribution table.

Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing postgraduate research. He has the following probability distribution:

Define the random variable X.

Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing postgraduate research. He has the following probability distribution:

Find the probability that a physics major will do postgraduate research for four years. P(x = 4) = ________

Use the following information to answer the next five exercises: A physics professor wants to know what percent of physics majors will spend the next several years doing postgraduate research. He has the following probability distribution:

On average, how many years would you expect a physics major to spend doing postgraduate research?

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution:

In words, define the random variable X.

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution:

P(x < 4) = ________

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution:

What does the column P(x) sum to and why?

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next so that she can plan what classes to offer. Over the years, she has established the following probability distribution:

You are playing a game by drawing a card from a standard deck and replacing it. If the card is a face card, you win $30. If it is not a face card, you pay $2. There are 12 face cards in a deck of 52 cards. What is the expected value of playing the game?

Use the following information to answer the next eight exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agreed with a recent federal law that was passed.

Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law.

In words, define the random variable X.

Use the following information to answer the next eight exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agreed with a recent federal law that was passed.

Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law.

What values does the random variable X take on?

Use the following information to answer the next eight exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agreed with a recent federal law that was passed.

Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law.

On average (μ), how many would you expect to answer yes?

Use the following information to answer the next eight exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agreed with a recent federal law that was passed.

Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number who agreed with that law.

What is the probability that at most five of the freshmen reply yes?

Use the following information to answer the next six exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agree with a recent law that was passed. Suppose that you randomly select freshman from the study until you find one who replies yes. You are interested in the number of freshmen you must ask.

In words, define the random variable X.

Use the following information to answer the next six exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agree with a recent law that was passed. Suppose that you randomly select freshman from the study until you find one who replies yes. You are interested in the number of freshmen you must ask.

What values does the random variable X take on?

Use the following information to answer the next six exercises: Researchers collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the United States. Of those students, 71.3 percent replied that, yes, they agree with a recent law that was passed. Suppose that you randomly select freshman from the study until you find one who replies yes. You are interested in the number of freshmen you must ask.

On average (μ), how many freshmen would you expect to have to ask until you found one who replies yes?

Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.

In words, define the random variable X.

Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.

What values does X take on?

Use the following information to answer the next five exercises: Suppose that a group of statistics students is divided into two groups: business majors and non-business majors. There are 16 business majors in the group and seven non-business majors in the group. A random sample of nine students is taken. We are interested in the number of business majors in the sample.

On average (μ), how many would you expect to be business majors?

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.

What values does X take on?

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.

What is the probability of getting 35 customers in the first four hours? Assume the store is open 12 hours each day.

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.

What is the probability that the store will have fewer than 12 customers in the first two hours?

Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.

Assume the event occurs independently in any given day. In words, define the random variable X.

Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.

What values does X take on?

Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.

Is it likely that there will be no teens killed from motor vehicle injuries on any given day in the United States? Justify your answer numerically.

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.

a. Find the expected value for this game (expected net gain or loss).

b. Explain what your calculations indicate about your long-term average profits and losses on this game.

c. Should you play this game to win money?

Complete the PDF and answer the questions.

a. Find the probability that x = 2.

b. Find the expected value.

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

a. Construct a PDF for each investment.

b. Find the expected value for each investment.

c. Which is the safest investment? Why do you think so?

d. Which is the riskiest investment? Why do you think so?

e. Which investment has the highest expected return, on average?

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 4.34.

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

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?

a. Yes, I expect to come out ahead in money.

b. No, I expect to come out behind in money.

c. It doesn’t matter. I expect to break even.

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

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 percent. 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.

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 percent. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu.

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

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 percent. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually 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 4.37. There is a five-video limit per customer at this store, so nobody ever rents more than five DVDs.

a. Describe the random variable X in words.

b. Find the probability that a customer rents three DVDs.

c. Find the probability that a customer rents at least four DVDs.

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

Use the following information to answer the next three exercises: The probability that a local hockey team 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.

What is the expected number of wins for that upcoming month?

a. 1.67

b. 12

c. $\frac{382}{1043}$

d. 4.43

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

What is the probability that the team wins at least five games in that upcoming month

a. .3694

b. .5266

c. .4734

d. .2305

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

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 percent of the questions correctly.

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

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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. On average, how many schools would you expect to offer such courses?

e. Find the probability that at most 10 offer such courses.

f. 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.

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

At the Fencing Center, 60 percent 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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many are expected to not to use the foil as their main weapon?

e. Find the probability that six do not use the foil as their main weapon.

f. Based on numerical values, would you be surprised if all 25 did not use foil as their main weapon? Justify your answer numerically.

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

The chance of an IRS audit for a tax return reporting more than $25,000 in income is about 2 percent 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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many audits are expected in a 20-year period?

e. Find the probability that a person is not audited at all.

f. Find the probability that a person is audited more than twice.

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

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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. List the values that Y may take on. Then, construct one PDF table that includes both X and Y and their probabilities.

e. Calculate the average expected matches over the long run of playing this game for the player.

f. Calculate the average expected earnings over the long run of playing this game for the player.

g. Determine who has the advantage, the player or the house.

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

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

a. Sketch a graph of the probability distribution of X.

b. Using the formulas, calculate the (i) mean and (ii) standard deviation of X.

c. Find the probability that more than five people in the sample are literate. Is it more likely that three people or four people are literate?

Suppose that the probability that an adult in America will watch the Super Bowl is 40 percent. 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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many adults in America do you expect to survey until you find one who will watch the Super Bowl?

e. Find the probability that you must ask seven people.

f. Find the probability that you must ask three or four people.

In one of its spring catalogs, a retailer 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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many pages do you expect to advertise footwear on them?

e. Is it probable that all 20 will advertise footwear on them? Why or why not?

f. What is the probability that fewer than 10 will advertise footwear on them?

g. 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.

h. What is the probability that you only need to survey at most three pages in order to find one that advertises footwear on it?

i. How many pages do you expect to need to survey in order to find one that advertises footwear?

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

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

a. Describe the distribution of X.

b. Find the (i) mean and (ii) standard deviation of X.

c. What is the probability that you must ask 10 people to find one person without a social networking site?

d. What is the probability that you must ask 20 people to find one person without a social networking site?

e. What is the probability that you must ask at most five people?

In one of its spring catalogs, a retailer 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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many pages do you expect to advertise footwear on them?

e. Calculate the standard deviation.

Suppose that nine Massachusetts athletes are scheduled to appear at a charity benefit. The nine are randomly chosen from eight volunteers from the local basketball team and four volunteers from the local football team. We are interested in the number of football players picked.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. Are you choosing the nine athletes with or without replacement?

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.

a. Find the mean and standard deviation of X.

b. What is the probability that the office receives at most six calls at noon on Monday?

c. 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?

d. What is the probability that the office receives more than eight calls at noon?

A manufacturer of decorative string lights knows that 3 percent of its bulbs are defective. Using both the binomial and Poisson distributions, find the probability that a string of 100 lights contains at most four defective bulbs.

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

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. Find the probability that she has no children.

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

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

The chance of having an extra fortune in a fortune cookie is about 3 percent. 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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many cookies do we expect to have an extra fortune?

e. Find the probability that none of the cookies have an extra fortune.

f. Find the probability that more than three have an extra fortune.

g. As n increases, what happens involving the probabilities using the two distributions? Explain in complete sentences.

The chance of an IRS audit for a tax return reporting more than $25,000 in income is about 2 percent 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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many are expected to be audited?

e. Find the probability that no one was audited.

f. Find the probability that at least three were audited.

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

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. On average, how many pieces of eggshell do you expect to be in the cake?

e. What is the probability that there will not be any pieces of eggshell in the cake?

f. 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 eggshell in any of the cakes?

g. 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 10. We are interested in the number of times her cats wake her up each week.

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

a. .5000

b. .9329

c. .0378

d. .0671

Use a programmable calculator to simulate a binomial distribution.

a. How would you use the randInt function to simulate the number of successes in five trials of an experiment with two outcomes, each of which has a .5 probability of occurring?

b. Use the randInt function to simulate 10 observations of the random variable in Part A.

c. Find the sample mean and sample standard deviation.

d. Compare the sample mean and sample standard deviation to the theoretical mean and the theoretical standard deviation.