AP Statistics - PRO: Probability

A chess match is played with a series of games. The winner of a game receives 1 point, while a drawn game gives each player \(\dfrac{1}{2}\) point. A player who scores 2 points before his or her opponent wins the match. Suppose Player A is the stronger player and when playing Player B will win with a probability of 0.6 and will draw with a probability 0.3. The outcome of any game is independent of the outcome of any other game. Player A and Player B begin a match. (a) What is the probability Player A wins the match in 2 games? (b) What is the probability Player A wins the match after exactly 3 games? (c) What is the probability that the first game was a draw given that Player A wins the match in exactly 3 games? (d) What is the probability Player A wins the second game given that he or she draws the first game?

You are asked to choose between two envelopes, one of which has twice as much money as the other. You arbitrarily pick one, open it, and find \$2. You are then given the chance to switch envelopes. You reason that the other envelope has either \$1 or \$4, each with a probability of 0.5. Applying your understanding of expected value, you calculate 0.5(\(1) + 0.5(\)4) = \$2.50 and conclude that you should switch envelopes. Comment on this reasoning.

The probability that Anthony Rizzo of the Chicago Cubs hits a homer on any given at bat is 0.06, and we assume each at bat is independent. (a) What is the probability the next homer will be on his fifth at bat? (b) What is the probability he hits exactly one homer in five at bats? (c) What is the expected number of homers in every 10 at bats? (d) What is the expected number of at bats until the next homer?

The weights of babies born to nonsmokers have a normal distribution with a mean of 7.0 pounds and a standard deviation of 0.8 pounds. Babies are considered low birth weight (LBW) if they weigh less than 5.5 pounds. Note that 5 percent of babies born to smokers are LBW and that 16 percent of pregnant women are smokers. (a) What is the probability that a nonsmoker will have an LBW baby? (b) What is the probability a baby is LBW? (c) Given that a baby is LBW, what is the probability the baby was born to a nonsmoker?

An online business charges shipping and handling (S&H) fees based on the sizes of the sales. 30 percent of the company's sales are under \$10, 40 percent are between \$10 and \$25, 20 percent are between \$25 and \$50, 5 percent are between \$50 and \$100, and 5 percent are over \$100. The charges are as follows, and note that S&H is free for orders over $100!

Sale $()	<10	10–25	25–50	50–100	>100
S&H Charge ()$	2.50	5.00	7.50	10.00	0.00

(a) What is the mean S&H charge? Show your work. (b) What is the median S&H charge? In other words, what charge \(C\) has the property that \(P(x \geq C) \geq 0.5\) and \(P(x \leq C) \geq 0.5\)? Explain.

Die A has three 5s, two 3s, and one 1 on its six faces. Die B has two 2s and four 4s on its six faces. Both dice are fair. Each player simultaneously rolls one of the dice, and the winner is the player with the higher number showing. (a) If you want to win, would you rather roll die A or die B? Explain. (b) If the winner receives whatever shows on his/her winning die, what is the expected value for one roll to each player? Explain.

The probability distribution for the number of days per week that college students get a good night's sleep is as follows.

Number of Days	0	1	2	3	4
Relative Frequency	0.12	0.41	0.25	0.15	0.07

(a) Calculate and give a brief interpretation of the mean of this probability distribution. (b) In a random sample of 10 college students, there are a total of 20 good nights of sleep. A new random sample of 50 students is planned. How do you expect the average number of good nights of sleep for this new sample to compare to that of the first sample? Explain. (c) Find the median of the above distribution, where the median \(M\) is defined to be a value such that \(P(x \geq M) \geq 0.5\) and \(P(x \leq M) \geq 0.5\).

A school must choose between two snow removal services. Service A charges \$2500 yearly plus \$2,000 for every month with over three snowfalls necessitating their services. Service B charges \$450 per snow removal. Relevant probabilities are shown in the following tables.

Month	Oct	Nov	Dec	Jan	Feb	Mar	Apr
P(> 3 snowfall services)	0.01	0.02	0.18	0.28	0.22	0.13	0.02
Annual Number of Snow Services	6	7	8	9	10	11	12
Probability	0.05	0.10	0.20	0.25	0.20	0.15	0.05

Which service should the school use to minimize expected cost? Justify your answer.

Suppose a pregnancy test correctly tests positive for 98 percent of pregnant women but also gives a false positive reading for 3 percent of women who are not pregnant. Suppose 5 percent of women who purchase this over-the-counter test are actually pregnant. (a) What is the probability a woman tests positive? (b) If a woman tests positive, what is the probability she is pregnant? (c) If two women purchase the test and both test positive, what is the probability that exactly one of the two is pregnant?

One model of a popular tablet computer has a life expectancy that is roughly normally distributed with a mean of 24 months and a standard deviation of 3 months. (a) The warranty will repair any computer failing within 18 months free of charge. The company's contracted cost to repair computers is \$350. What is the expected value of the cost to the company per computer? (b) Suppose the company decides it is willing to extend the warranty to an additional 5 percent of the computers. What warranty should the company offer?

Simple random samples of younger adults and older adults in a large city were surveyed as to the number of apps on their smart phones. The data are summarized in the following table. Is there evidence that either of these samples were drawn from populations with roughly normal distributions? Explain.

Simple random samples of younger adults and older adults in a large city were surveyed as to the amount of credit card debt, and the data are summarized in the following table. Assume that both samples are drawn from roughly normally distributed populations. Would a greater percentage of younger or of older adults more likely be able to pay off their credit debt with \$8,000? Explain.

Medical scientists applying to the CDC for grants are awarded scores for their grant applications. These scores follow a roughly normal distribution. All applicants with scores more than 1.5 standard deviations above the mean are awarded grants, and all applicants with scores that are outliers on the high end are awarded additional lab equipment in addition to the grants. (a) What is the probability a randomly selected scientist receives the grant? (b) What is the probability that at least one out of three randomly chosen scientists receives the grant? (c) What is the probability a randomly selected scientist receives the grant and the additional lab equipment? (d) What is the probability a randomly selected scientist receives the grant but not the additional lab equipment?

New Caledonian crows are among the smartest animals on this planet. An experiment testing the length of time it takes these crows to solve a particular puzzle (and receive a food reward) finds that 5 percent of the crows fail. Of those that do solve the puzzle, their times are roughly normally distributed with a mean of 1.8 minutes and a standard deviation of 0.4 minutes. (a) What is the probability a randomly selected crow solves the puzzle in under 1.5 minutes? (b) What is the probability a randomly selected crow has not solved the puzzle in 2.0 minutes?

MLB signing bonuses for different position players with respective probabilities are given in the table below.

Position	Pitcher	Infielder	Outfielder
Bonus
$2000000	0.30	0.30	0.20

\$5,000,000 0.10 0.05 0.05 (a) What is the probability a given bonus was for a pitcher? (b) What is the expected value for a bonus? (c) Are position and bonus independent? Explain.

A cereal box is advertised to hold 16 ounces of corn flakes. Suppose the amounts per box are roughly normally distributed with a standard deviation of 0.05 ounces and a mean \(\mu\) that the producer can set. An inspector randomly samples two boxes, and the company is fined \$10,000 for each box found to be under 16 ounces. (a) If the company is willing to accept an expected value of \$125 for the fine, what probability \(p\) should it be willing to accept where \(p\) is the probability that a box is under 16 ounces? (b) What should \(\mu\) be set at if the company is willing to accept an expected value of \$125 for the fine?

The weights of individual apples are approximately normally distributed with a mean of 8 ounces and a standard deviation of 0.5 ounces. The weights of individual oranges are approximately normally distributed with a mean of 6 ounces and a standard deviation of 0.4 ounces. The weights of individual pieces of fruit are independent. (a) What is the distribution of the total fruit weight of fruit gift boxes containing 6 randomly selected apples and 6 randomly selected oranges? (b) The gift boxes are advertised as containing 5 pounds of fruit. What is the probability that a gift box contains at least 5 pounds of fruit? (c) An empty gift box weighs exactly 12 ounces. What is the distribution of total weights (box plus fruit) of this gift offering?

The standard triathlon is a 1.5 km swim, 40 km bike ride, and 10 km run. In a random sampling of recent competitions, the mean and SD of the participants' times for each event were as shown in the table.

	Swim	Bike Ride	Run
Mean (minutes)	30	90	70
SD (minutes)	5	10	10

Assume the times for the three legs of the race are each roughly normally distributed and independent. (a) What is the distribution of total times to complete the triathlon? (b) What is the probability that a participant will complete the triathlon in less than 180 minutes?

An adult amusement park ride has a carrying capacity of 1,650 lb. Suppose the men at the amusement park have a mean weight of 175 lb with a standard deviation of 15 lb. Suppose the women have a mean weight of 130 lb with a standard deviation of 10 lb. Assume both weight distributions are roughly normal. (a) What are the mean \(\mu_{\text{SUM}}\) and standard deviation \(\sigma_{\text{SUM}}\) of the combined weight of 6 men and 4 women, assuming all weights are independent? (b) What is the probability that the 6 men and 4 women will overload the ride?

Suppose that women's times for the 200-meter sprint have a roughly normal distribution with a mean of 25.2 seconds and a standard deviation of 1.2 seconds. Suppose that men's times have a roughly normal distribution with a mean of 22.8 seconds and a standard deviation of 0.9 seconds. A male and a female sprinter are picked at random. Assume their times are independent. (a) What is the probability the sum of their sprints is over 50 seconds? (b) What is the probability that the man sprinted faster than the woman?

Eddie Feigner, the world's best softball pitcher, routinely threw an underhand softball at over 100 mph. Suppose his fastball speeds were roughly normally distributed with a mean of 98.3 mph and a standard deviation of 3.2 mph. (a) Would it have been very unusual for him to throw a 102 mph fastball? Explain. (b) Would it have been very unusual for him to average 102 mph over 35 random fastballs? Explain.

Suppose the number of minutes per day high school students spend eating lunch in the school cafeteria is roughly normally distributed with a mean of 19 minutes. (a) Which is more likely: a simple random sample (SRS) of 25 students eating lunch an average of less than 18 minutes per day or an SRS of 75 students eating lunch an average of less than 18 minutes per day? Explain. (b) Suppose the sampling distribution of \(\overline{x}\) for samples of size 100 has a standard deviation of 0.8 minutes. What is the probability of an SRS of 100 students eating lunch an average of more than 21 minutes? (c) Suppose the original population is not roughly normal but, rather, is skewed right (toward the higher values). How would your calculation in (b) change?