196 CHAPTER 4 Probability 13.Simulating the Monty Hall Problem A problem that once attracted much attention is the Monty Hall problem, based on the old television game show Let’s Make a Deal, hosted by Monty Hall. Suppose you are a contestant who has selected one of three doors after being told that two of them conceal nothing, but that a new red Corvette is behind one of the three. Next, the host opens one of the doors you didn’t select and shows that there is nothing behind it. He then offers you the choice of sticking with your first selection or switching to the other unopened door. Should you stick with your first choice or should you switch? Develop a simulation of this game and determine whether you should stick or switch. (According to Chance magazine, business schools at such institutions as Harvard and Stanford use this problem to help students deal with decision making.) 14.Simulating Birthdays a. Develop a simulation for finding the probability that when 50 people are randomly selected, at least 2 of them have the same birth date. Describe the simulation and estimate the probability. b. Develop a simulation for finding the probability that when 50 people are randomly selected, at least 3 of them have the same birth date. Describe the simulation and estimate the probability. 15.Genetics: Simulating Population Control A classical probability problem involves a king who wanted to increase the proportion of women by decreeing that after a mother gives birth to a son, she is prohibited from having any more children. The king reasons that some families will have just one boy, whereas other families will have a few girls and one boy, so the proportion of girls will be increased. Conduct a simulation to determine whether his reasoning is correct, and to determine whether the proportion of girls will increase. 4-5 Beyond the Basics 1. ESP A psychologist tells you that in an ESP (extrasensory perception) experiment, there is a 20% chance of answering a question correctly. What is the probability of answering a question correctly? 2.Standard Tests Standard tests, such as the SAT or ACT or MCAT, tend to make extensive use of multiple-choice questions because they are easy to grade using software. If one such multiple choice question has possible correct answers of a, b, c, d, e, what is the probability of a wrong answer if the answer is a random guess? 3. Birthday If a day of a year (not a leap year) is randomly selected, what is the probability it is the author’s birthday? 4.Online Courses Based on data from a survey sponsored by Sallie Mae, 10% of undergraduate students take online courses only. If two undergraduate students are randomly selected, what is the probability that they both take online courses only? 5.Subjective Probability Estimate the probability that the next time you watch a TV news report, it includes a story about a plane crash. In Exercises 6–10, use the following results from tests of an experiment to test the effectiveness of an experimental vaccine for children (based on data from USA Today). Express all probabilities in decimal form. Developed Flu Did Not Develop Flu Vaccine Treatment 14 1056 Placebo 95 437 6. If 1 of the 1602 subjects is randomly selected, find the probability of getting 1 that developed flu. Chapter Quick Quiz
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