Text book: Statistics: Concepts and Controversies.

Complete 8 of the following:

Chapter 17: Exercise 17.22; page 422

Playing Pick 4. The Pick 4 games in many state lotteries announce a four-digit winning number each day. The winning number is essentially a four-digit group from a table of random digits. You win if your choice matches the winning digits, in exact order. The winnings are divided among all players who matched the winning digits. That suggests a way to get an edge.

(a) The winning number might be, for example, either 2873 or 9999. Explain why these two outcomes have exactly the same probability. (It is 1 in 10,000.)

(b) If you asked many people which outcome is more likely to be the randomly chosen winning number, most would favor one of them. Use the information in this chapter to say which one and to explain why. If you choose a number that people think is unlikely, you have the same chance to win, but you will win a larger amount because few other people will choose your number.

Chapter 17: Exercise 17.32; page 423

17.32 What probability doesn't say.The probability of a head in tossing a coin is 1-in-2. This means that as we make more tosses, theproportionof heads will eventually get close to 0.5. It does not mean that thecountof heads will get close to one-half the number of tosses. To see why, imagine that the proportion of heads is 0.49 in 100 tosses, 0.493 in 1000 tosses, 0.4969 in 10,000 tosses, and 0.49926 in 100,000 tosses of a coin. How many

heads came up in each set of tosses? How close is the number of heads to half the number of tosses?

Chapter 18: Exercise 18.10; page 436

Land in Canada. Choose an acre of land in Canada at random. The probability is 0.45 that it is forest and 0.03 that it is pasture.

(a) What is the probability that the acre chosen is not forested?

(b) What is the probability that it is either forest or pasture?

(c) What is the probability that a randomly chosen acre in Canada is something other than forest or pasture?

Chapter 19: Exercise 19.19; page 457 (use Table A on p. 641 of the textbook, starting at line 130, to answer this question)

A more realistic probability model for Elaine's attempts to pass an exam in the previous exercise is as follows. On the first try she has probability 0.2 of passing. If she fails on the first try, her probability on the second try increases to 0.3 because she learned something from her first attempt. If she fails on two attempts, the probability of passing on a third attempt is 0.4. She will stop as soon as she passes. The course rules force her to stop after three attempts in any case.

(a) Make a tree diagram of Elaine's progress. Notice that she has different probabilities of passing on each successive try.

(b) Explain how to simulate one repetition of Elaine's tries at the exam. (c) Simulate 50 repetitions and estimate the probability that Elaine

eventually passes the exam. Use Table A, starting at line 130.

Chapter 20: Exercise 20.8; page 471

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?

Chapter 20: Exercise 20.28; page 474

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.

(a) After two bets, how much does he have of his original dollar bet? (b) After three bets, how much does he have of his original dollar bet?

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.

Part III Review: Exercise III.6; page 480

Grades in an economics course. Choose a student at random from the course described in Exercise III.3 and observe what grade that student earns (with A = 4, A = 3.7, B+ = 3.3, B = 3.0, B = 2.7, C+ = 2.3, C = 2.0, C = 1.7, D+ = 1.3, D = 1.0, D = 0.7, and F = 0.0).

(a) What is the expected grade of a randomly chosen student?

(b) The expected grade is not one of the 12 grades possible for one student. Explain why your result nevertheless makes sense as an expected value.

Part III Review: Exercise III.22; page 482

Predicting the winner. There are 14 teams in the Big Ten athletic conference. Here's one set of personal probabilities for next year's basketball champion: Michigan State has probability 0.3 of winning. Illinois, Iowa, Minnesota, Nebraska, Northwestern, Penn State, and Rutgers have no chance. That leaves six teams. Indiana, Michigan, Purdue, Ohio State, and Wisconsin also all have the same probability of winning, but that probability is one-half that of Maryland. What probability does each of the 14 teams have?