20.8 The numbers racket. Pick 3 lotteries (Example 1) copy the numbers racket, an illegal gambling operation common in the poorer areas of large cities. States usually justify their lotteries by donating a portion of the proceeds to education. One version of a numbers racket operation works as follows. You choose any one of the 1000 three-digit numbers 000 to 999 and pay your local numbers runner $1 to enter your bet. Each day, one three-digit number is chosen at random and pays off $600. What is the expected value of a bet on the numbers? Is the numbers racket more or less favorable to gamblers than the Pick 3 game in Example 1?

20.9 Pick 4. The Tri-State Daily Numbers Pick 4 is much like the Pick 3 game of Example 1. Winning numbers for both are reported on television and in local newspapers. You pay $0.50 and pick a four-digit number. The state chooses a four-digit number at random and pays you $2500 if your number is chosen. What are the expected winnings from a $0.50 Pick 4 wager?

20.10 More Pick 4. Just as with Pick 3 (Example 2), you can make more elaborate bets in Pick 4. In the $1 StraightBox (24-way) bet, if you choose 1234 you win $2604 if the randomly chosen winning number is 1234, and you win $104 if the winning number has the digits 1, 2, 3, and 4 in any other order (there are 15 such other orders). What is the expected amount you win?

20.11 More roulette. An American roulette wheel has 38 slots, of which 18 are black, 18 are red, and 2 are green. When the wheel is spun, the ball is equally likely to come to rest in any of the slots. Gamblers bet on roulette by placing chips on a table that lays out the numbers and colors of the 38 slots in the roulette wheel. The red and black slots are arranged on the table in three columns of 12 slots each. A $1 column bet wins $3 if the ball lands in one of the 12 slots in that column. What is the expected amount such a bet wins? If you did the Case Study Evaluated, is a column bet more or less favorable to a gambler than a bet on red or black (see Question 1 in the Case Study Evaluated on page 473)?

20.12 Making decisions. The psychologist Amos Tversky did many studies of our perception of chance behavior. In its obituary of Tversky, the New York Times cited the following example.

20.13 Making decisions. A six-sided die has two green and four red faces and is balanced so that each face is equally likely to come up. You must choose one of the following three sequences of colors:

Now start rolling the die. You will win $25 if the first rolls give the sequence you chose.

20.14 Estimating sales. Gain Communications sells aircraft communications units. Next year’s sales depend on market conditions that cannot be predicted exactly. Gain follows the modern practice of using probability estimates of sales. The sales manager estimates next year’s sales as follows:

These are personal probabilities that express the informed opinion of the sales manager. What is the sales manager’s expected value of next year’s sales?

20.15 Keno. Keno is a popular game in casinos. Balls numbered 1 to 80 are tumbled in a machine as the bets are placed, then 20 of the balls are chosen at random. Players select numbers by marking a card. Here are two of the simpler Keno bets. Give the expected winnings for each.

20.16 Rolling two dice. Example 2 of Chapter 18 (page 430) gives a probability model for rolling two casino dice and recording the number of spots on each of the two up-faces. That example also shows how to find the probability that the total number of spots showing is five. Follow that method to give a probability model for the total number of spots. The possible outcomes are 2, 3, 4, . . . , 12. Then use the probabilities to find the expected value of the total.

20.17 The Asian stochastic beetle. We met this insect in Exercise 19.21 (page 460). Females have this probability model for their number of female offspring:

20.18 An expected rip-off? A “psychic” runs the following ad in a magazine:

Expecting a baby? Renowned psychic will tell you the sex of the unborn child from any photograph of the mother. Cost, $20. Money-back guarantee.

This may be a profitable con game. Suppose that the psychic simply replies “boy” to all inquiries. In the worst case, everyone who has a girl will ask for her money back. Find the expected value of the psychic’s profit by filling in the table below.

20.19 The Asian stochastic beetle again. In Exercise 20.17, you found the expected number of female offspring of the Asian stochastic beetle. Simulate the offspring of 100 beetles and find the mean number of offspring for these 100 beetles. Compare this mean with the expected value from Exercise 20.17. (The law of large numbers says that the mean will be very close to the expected value if we simulate enough beetles.)

20.20 Life insurance. You might sell insurance to a 21-year-old friend. The probability that a man aged 21 will die in the next year is about 0.0008. You decide to charge $2000 for a policy that will pay $1,000,000 if your friend dies.

20.21 Household size. The Census Bureau gives this distribution for the number of people in American households in 2011:

(Note: In this table, 7 actually represents households of size 7 or greater. But for purposes of this exercise, assume that it means only households of size exactly 7.)

20.22 Course grades. The distribution of grades in a large accelerated introductory statistics course is as follows:

To calculate student grade point averages, grades are expressed in a numerical scale with A = 4, B = 3, and so on down to F = 0.

20.23 We really want a girl. Example 4 estimates the expected number of children a couple will have if they keep going until they get a girl or until they have three children. Suppose that they set no limit on the number of children but just keep going until they get a girl. Their expected number of children must now be higher than in Example 4. How would you simulate such a couple’s children? Simulate 25 repetitions. What is your estimate of the expected number of children?

20.24 Play this game, please. OK, friends, we’ve got a little deal for you. We have a fair coin (heads and tails each have probability 1-in-2). Toss it twice. If two heads come up, you win right there. If you get any result other than two heads, we’ll give you another chance: toss the coin twice more, and if you get two heads, you win. (Of course, if you fail to get two heads on the second try, we win.) Pay us a dollar to play. If you win, we’ll give you your dollar back plus another dollar.

20.25 A multiple-choice exam. Charlene takes a quiz with 10 multiple-choice questions, each with five answer choices. If she just guesses independently at each question, she has probability 0.20 of guessing right on each. Use simulation to estimate Charlene’s expected number of correct answers. (Simulate 20 repetitions.)

20.26 Repeating an exam. Exercise 19.19 (page 460) gives a model for up to three attempts at an exam in a self-paced course. In that exercise, you simulated 50 repetitions to estimate Elaine’s probability of passing the exam. Use those simulations (or do 50 new repetitions) to estimate the expected number of tries Elaine will make.

20.27 A common expected value. Here is a common setting that we simulated in Chapter 19: there are a fixed number of independent trials with the same two outcomes and the same probabilities on each trial. Tossing a coin, shooting basketball free throws, and observing the sex of newborn babies are all examples of this setting. Call the outcomes “hit” and “miss.” We can see what the expected number of hits should be. If Stephen Curry shoots 12 three-point shots and has probability 0.44 of making each one, the expected number of hits is 44% of 12, or 5.28. By the same reasoning, if we have n trials with probability p of a hit on each trial, the expected number of hits is np. This fact can be proved mathematically. Can we verify it by simulation?

Simulate 10 tosses of a fair coin 50 times. (To do this quickly, use the first 10 digits in each of the 50 rows of Table A, with odd digits meaning a head and even digits a tail.) What is the expected number of heads by the np formula? What is the mean number of heads in your 50 repetitions?

20.28 Casino winnings. What is a secret, at least to naive gamblers, is that in the real world, a casino does much better than expected values suggest. In fact, casinos keep a bit over 20% of the money gamblers spend on roulette chips. That’s because players who win keep on playing. Think of a player who gets back exactly 95% of each dollar bet. After one bet, he has 95 cents.

Notice that the longer he keeps recycling his original dollar, the more of it the casino keeps. Real gamblers don’t get a fixed percentage back on each bet, but even the luckiest will lose his stake if he plays long enough. The casino keeps 5.3 cents of every dollar bet but 20 cents of every dollar that walks in the door.