Problem 19
Multiple choice. You take a quiz with 6 multiple choice questions. After you studied, you estimated that you would have about an \(80 \%\) chance of getting any individual question right. What are your chances of getting them all right? Use at least 20 trials.
Problem 21
Beat the lottery. Many states run lotteries to raise money. A Web site advertises that it knows "how to increase YOUR chances of Winning the Lottery." They offer several systems and criticize others as foolish. One system is called Lucky Numbers. People who play the Lucky Numbers system just pick a "lucky" number to play, but maybe some numbers are luckier than others. Let's use a simulation to see how well this system works. To make the situation manageable, simulate a simple lottery in which a single digit from 0 to 9 is selected as the winning number. Pick a single value to bet, such as 1 , and keep playing it over and over. You'll want to run at least 100 trials. (If you can program the simulations on a computer, run several hundred. Or generalize the questions to a lottery that chooses two- or three-digit numbers-for which you'll need thousands of trials.) a) What proportion of the time do you expect to win? b) Would you expect better results if you picked a "luckier" number, such as \(7 ?\) (Try it if you don't know.) Explain.
Problem 25
Driving test. You are about to take the road test for your driver's license. You hear that only \(34 \%\) of candidates pass the test the first time, but the percentage rises to \(72 \%\) on subsequent retests. Estimate the average number of tests drivers take in order to get a license. Your simulation should use at least 20 runs.
Problem 27
Basketball strategy. Late in a basketball game, the team that is behind often fouls someone in an attempt to get the ball back. Usually the opposing player will get to shoot foul shots "one and one," meaning he gets a shot, and then a second shot only if he makes the first one. Suppose the opposing player has made \(72 \%\) of his foul shots this season. Estimate the number of points he will score in a one-and-one situation.
Problem 28
Blood donors. A person with type O-positive blood can receive blood only from other type \(\mathrm{O}\) donors. About \(44 \%\) of the U.S. population has type \(\mathrm{O}\) blood. At a blood drive, how many potential donors do you expect to examine in order to get three units of type \(\mathrm{O}\) blood?
Problem 29
Free groceries. To attract shoppers, a supermarket runs a weekly contest that involves "scratch-off" cards. With each purchase, customers get a card with a black spot obscuring a message. When the spot is scratched away, most of the cards simply say, "Sorry-please try again." But during the week, 100 customers will get cards that make them eligible for a drawing for free groceries. Ten of the cards say they may be worth \(\$ 200,10\) others say \(\$ 100,20\) may be worth \(\$ 50\), and the rest could be worth \(\$ 20\). To register those cards, customers write their names on them and put them in a barrel at the front of the store. At the end of the week the store manager draws cards at random, awarding the lucky customers free groceries in the amount specified on their card. The drawings continue until the store has given away more than \(\$ 500\) of free groceries. Estimate the average number of winners each week.
Problem 30
Find the ace. A new electronics store holds a contest to attract shoppers. Once an hour someone in the store is chosen at random to play the Music Game. Here's how it works: An ace and four other cards are shuffled and placed face down on a table. The customer gets to turn cards over one at a time, looking for the ace. The person wins \(\$ 100\) worth of free CDs or DVDs if the ace is the first card, \(\$ 50\) if it is the second card, and \(\$ 20, \$ 10\), or \(\$ 5\) if it is the third, fourth, or fifth card chosen. What is the average dollar amount of music the store will give away?
Problem 31
The family. Many couples want to have both a boy and a girl. If they decide to continue to have children until they have one child of each sex, what would the average family size be? Assume that boys and girls are equally likely.
Problem 32
A bigger family. Suppose a couple will continue having children until they have at least two children of each sex (two boys and two girls). How many children might they expect to have?
Problem 36
The World Series. The World Series ends when a team wins 4 games. Suppose that sports analysts consider one team a bit stronger, with a \(55 \%\) chance to win any individual game. Estimate the likelihood that the underdog wins the series.
