19.8 Which party does it better? An opinion poll selects adult Americans at random and asks them, “Which political party, Democratic or Republican, do you think is better able to keep the U.S. prosperous?” Explain carefully how you would assign digits from Table A to simulate the response of one person in each of the following situations.

19.9 A small opinion poll. Suppose that 80% of a university’s students favor increasing the number of police patrolling campus at night. You ask 10 students chosen at random. What is the probability that all 10 favor increasing the number of police patrolling campus at night?

19.10 Basic simulation. Use Table A to simulate the responses of 10 independently chosen adults in each of the four situations of Exercise 19.8.

19.11 Simulating an opinion poll. A Gallup Poll on Presidents Day 2011 interviewed a random sample of 1015 adult Americans. Those in the sample were asked which U.S. president they regarded as the greatest. The poll showed that about 20% of adult Americans regarded Ronald Reagan as the greatest U.S. president. Suppose that this is exactly true. Choosing an adult American at random then has probability 0.2 of getting one who would identify Ronald Reagan as the greatest president. If we interview adult Americans separately, we can assume that their responses are independent. We want to know the probability that a simple random sample of 100 adult Americans will contain at least 25 who say that Ronald Reagan is the greatest U.S. president. Explain carefully how to do this simulation and simulate one repetition of the poll using line 112 of Table A. How many of the 100 adult Americans said that Ronald Reagan is the greatest U.S. president? Explain how you would estimate the probability by simulating many repetitions.

19.12 Course grades. Choose a student at random from all who took the large accelerated introductory statistics at Hudson River College in the last 10 years. The probabilities for the student’s grade are

19.13 The demands of college. Select a first-year college student at random and ask how difficult it was for him or her to adjust to the academic demands of college. Probabilities for the outcomes are

19.14 More on course grades. In Exercise 19.12, you explained how to simulate the grade of a randomly chosen student who took the accelerated statistics course in the last 10 years. Suppose you select five students at random who took the course in the last 10 years. Use simulation to estimate the probability that all five got a C or better in the course. (Simulate 20 repetitions and assume the student grades are independent of each other.)

19.15 More on the demands of college. In Exercise 19.13, you explained how to simulate the response of a randomly chosen first-year college student to the question of how difficult it was to adjust to the academic demands of college. The Random Committee decides to choose eight students at random and provide them with training on adjusting to the academic demands of college. What is the probability that at least five of the eight students chosen were among those who reported that it was very difficult or somewhat difficult to adjust to the academic demands of college? Simulate 10 repetitions of the committee’s choices to estimate this probability.

19.16 Curry’s three-point shooting. The basketball player Stephen Curry makes about 44% of his three-point shots over an entire season. Take his probability of a success to be 0.44 on each shot. Using line 122 of Table A, simulate 25 repetitions of his performance in a game in which he shoots 10 three-point shots.

19.17 Elena’s free throws. Elena Delle Donne of the WNBA Chicago Sky makes 94% of her free throws. In an important game, she shoots a pair of free throws shots late in the game and misses both. The fans think she was nervous, but the misses may simply be chance. Let’s shed some light by estimating a probability.

19.18 Repeating an exam. Elaine is enrolled in a self-paced course that allows three attempts to pass an examination on the material. She does not study and has probability 2-in-10 of passing on any one attempt by luck. What is Elaine’s probability of passing in three attempts? (Assume the attempts are independent because she takes a different examination on each attempt.)

19.19 A better model for repeating an exam. 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.

19.20 Gambling in ancient Rome. Tossing four astragali was the most popular game of chance in Roman times. Many throws of a present-day sheep’s astragalus show that the approximate probability distribution for the four sides of the bone that can land uppermost are

The best throw of four astragali was the “Venus,” when all four uppermost sides were different.

19.21 The Asian stochastic beetle. We can use simulation to examine the fate of populations of living creatures. Consider the Asian stochastic beetle. Females of this insect have the following pattern of reproduction:

What will happen to the population of Asian stochastic beetles: will they increase rapidly, barely hold their own, or die out? It’s enough to look at the female beetles, as long as there are some males around.

19.22 Two warning systems. An airliner has two independent automatic systems that sound a warning if there is terrain ahead (that means the airplane is about to fly into a mountain). Neither system is perfect. System A signals in time with probability 0.9. System B does so with probability 0.8. The pilots are alerted if either system works.

19.23 Playing craps. The game of craps is played with two dice. The player rolls both dice and wins immediately if the outcome (the sum of the faces) is 7 or 11. If the outcome is 2, 3, or 12, the player loses immediately. If he rolls any other outcome, he continues to throw the dice until he either wins by repeating the first outcome or loses by rolling a 7.

19.24 The airport van. Your company operates a van service from the airport to downtown hotels. Each van carries seven passengers. Many passengers who reserve seats don’t show up—in fact, the probability is 0.2 that a randomly chosen passenger will fail to appear. Passengers’ behaviors are independent. If you allow nine reservations for each van, what is the probability that more than seven passengers will appear? Do a simulation to estimate this probability.

19.25 A multiple-choice exam. Matt has lots of experience taking multiple-choice exams without doing much studying. He is about to take a quiz that has 10 multiple-choice questions, each with four possible answers. Here is Matt’s personal probability model. He thinks that in 75% of questions he can eliminate one answer as obviously wrong; then he guesses from the remaining three. He then has probability 1-in-3 of guessing the right answer. For the other 25% of questions, he must guess from all four answers, with probability 1-in-4 of guessing correctly.

19.26 More on the airport van. Let’s continue the simulation of Exercise 19.24. You have a backup van, but it serves several stations. The probability that it is available to go to the airport at any one time is 0.6. You want to know the probability that some passengers with reservations will be left stranded because the first van is full and the backup van is not available. Draw a tree diagram with the first van (full or not) as the first stage and the backup (available or not) as the second stage. In Exercise 19.24, you simulated a number of repetitions of the first stage. Add simulations of the second stage whenever the first van is full. What is your estimate of the probability of stranded passengers?

19.27 The birthday problem. A famous example in probability theory shows that the probability that at least two people in a room have the same birthday is already greater than 1-in-2 when 23 people are in the room. The probability model is

To simulate birthdays, let each three-digit group in Table A stand for one person’s birth date. That is, 001 is January 1 and 365 is December 31. Ignore leap years and skip groups that don’t represent birth dates. Use line 139 of Table A to simulate birthdays of randomly chosen people until you hit the same date a second time. How many people did you look at to find two with the same birthday?

With a computer, you could easily repeat this simulation many times. You could find the probability that at least two out of 23 people have the same birthday, or you could find the expected number of people you must question to find two with the same birthday. These problems are a bit tricky to do by math, so they show the power of simulation.

19.28 The multiplication rule. Here is another basic rule of probability: if several events are independent, the probability that all of the events happen is the product of their individual probabilities. We know, for example, that a child has probability 0.49 of being a girl and probability 0.51 of being a boy and that the sexes of successive children are independent. So the probability that a couple’s two children are two girls is (0.49)(0.49) = 0.2401. You can use this multiplication rule to calculate the probability that we simulated in Example 4.