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

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have 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 4.1 using the data provided.

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have 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, in other words, how long new hires stay with the company. Over the years, they have 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 six 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 six 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 four exercises: Emery has music practice three days a week. They practice 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.

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% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% 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% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% 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% of the time, four events 25% of the time, three events 20% of the time, two events 10% of the time, one event 5% of the time, and no events 5% of the time.

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

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 eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time first-year students from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time first-year students from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.

In words, define the random variable X.

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time first-year students from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time first-year students from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.

What values does the random variable X take on?

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time first-year students from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time first-year students from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.

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

Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time first-year students from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time first-year students from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.

What is the probability that at most five of the first-year students reply “yes”?

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time first-year students from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select first-year students from the study until you find one who replies “yes.” You are interested in the number of first-year students you must ask.

In words, define the random variable X.

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time first-year students from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select first-year students from the study until you find one who replies “yes.” You are interested in the number of first-year students you must ask.

What values does the random variable X take on?

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time first-year students from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select first-year students from the study until you find one who replies “yes.” You are interested in the number of first-year students you must ask.

On average (μ), how many first-year students would you expect to have to ask until you found one who replies "yes?"

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 U.S. 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 U.S. 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 U.S. 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 U.S? Justify your answer numerically.

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.

a. In words, define the random variable X.

b. List the values that X may take on.

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

d. 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 Boston Celtics and four volunteers from the New England Patriots. We are interested in the number of Patriots picked.

a. In words, define the random variable X.

b. List the values that X may take on.

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

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 they truly have 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.

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 they truly have 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.

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 they truly have 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.

People visiting video game rental stores often rent more than one game at a time. The probability distribution for game rentals per customer at Game Stop is given Table 4.5. There is five-game limit per customer at this store, so nobody ever rents more than five video games.

a. Describe the random variable X in words.

b. Find the probability that a customer purchases three games.

c. Find the probability that a customer purchases at least four games.

d. Find the probability that a customer purchases at most two games.

Use the following information to answer the next two 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:

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 San Jose Sharks win at least five games in that upcoming month

a. 0.3694

b. 0.5266

c. 0.4734

d. 0.2305

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% 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 ten 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% 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 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.

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 their $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 their $1 bet, plus $2 profit. If all three dice show the number or object bet, the player gets back their $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. List the values that Y may take on. Then, construct one PDF table that includes both X and Y and their probabilities.

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

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

f. 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 that can read and write. The literacy rate in a certain country is 28.1%. Suppose you choose 15 people in a certain country 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 is 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%. 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, 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.

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 twenty will advertise footwear on them? Why or why not?

f. What is the probability that fewer than ten 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% 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?

According to a recent Pew Research poll, 75% 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 ten 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?

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 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 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. Find the probability that she has no children.

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

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

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

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

f. 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 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 answer the following questions.

a. In words, define the random variable X.

b. List the values that X may take on.

c. How many are expected to be audited?

d. Find the probability that no one was audited.

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

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.

a. In words, define the random variable X.

b. List the values that X may take on.

c. On average, how many pieces of egg shell do you expect to be in the cake?

d. What is the probability that there will not be any pieces of egg shell in the cake?

e. 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?

f. 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.

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

a. 0.5000

b. 0.9329

c. 0.0378

d. 0.0671