16.9: Chapter Exercises

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Chapter Exercises

Evaluate the following argument, considering concepts recently covered in class.

The burglary rate in Belleville, Kansas has always been very close to 1 robbery per 400 homes. But last year it ballooned up to 3.9 per 400. However, the Chief of Police quickly hired three additional policemen, and this year the burglary rate is back down to where it had been before last year’s increase. So, hiring more police is a good way to lower the burglary rate.

Children doing below average work in school who suddenly do well on an achievement test are often labeled underachievers. Sometimes they are, but what else might be going on?
Phil often find that he is disappointed when he returns to a restaurant that seemed outstanding on the first visit. He’s often tempted to conclude that the chefs got lazy over time or that the management quit working as hard as they had at the beginning. What do you think about his reasoning? What other explanations might there be for this result?
Wilbur reasons in the following way: If a method of contraception has a 6% failure rate, then we should expect the same probability of getting pregnant in 1 year of use as we would in 10 years of use. The chances are that 6% of the people using it will create a pregnancy.
If a student gets the highest grade in her class on the first examination in Critical Reasoning, what grade would you predict that she will get on the midterm? Justify your answer.
Most of you had to take the ACT or the SAT test; if you apply to graduate school you will have to take the GRE, and if you apply to Law School you will have to take the LSAT. These exams involve multiple choice questions. To discourage random guessing, many tests of this sort subtract points for wrong answers. Suppose that a correct answer is worth +1 point and that an incorrect answer on a question with 5 listed answers (a through e) is worth -1/4 point.
1. Find the expected value of a random guess.
2. Find the expected value of eliminating one answer and guessing between the remaining 4 possible answers.
3. Using your answers to (a) and (b), when would it be advisable to guess and why?
Suppose that you take a multiple-choice exam consisting of ten questions. Each question has four possible answers. The topic is one you know nothing about, and you are reduced to guessing. What is the probability that you will guess right on the first question? What is the probability that all ten of your guesses will be right (you know nothing about the subject matter, so you guess at random and so you can assume independence)? If one million people took the exam, how good are the chances that at least one person would get all the answers right simply by random guessing (don’t worry about assigning a number to this, but be as precise as you can, and justify your answer).
From a lecture on fire safety in the home: “One in ten Americans will experience some type of destructive fire this year. Now, I know that some of you can say that you have lived in your home for 25 years and never had any type of fire. To that, I would respond that you have been lucky…. But that only means that you are not moving farther away from a fire, but closer to one.” Evaluate the reasoning in this passage.
Diego makes an average of 35% of his basketball shots. After playing in a pick-up game in which he misses all six of the shots he takes, he argues that in the next game he will be hot because, having missed six already, he has the odds in his favor. Evaluate Diego’s reasoning.
Suppose that you built a computer that had 500 independent parts. And suppose that each part was 99% reliable when used the very first time. What are the chances that such a computer would work the very first time it was turned on?
In the previous chapter, we noted Laurie Anderson’s quip: “The chances of there being two bombs on a plane are very small. So, when I fly, I always take along a bomb.” We are now in a better position to analyze the bad reasoning involved. Do so.

Answers to Selected Exercises

“The chances of there being two bombs on a plane are very small, So, when I fly I always take along a bomb.” Assuming I am not in league with any terrorists, whether I bring a bomb has no effect on whether someone else also brings a bomb along on the flight. The two events are independent. But the joke treats them as though they were dependent (my bringing a bomb makes it less likely that others will). Hence it involves a subtle instance of the gambler’s fallacy.