1) Sometimes I hear students insisting that they don't know anyone who is gay. I have read that approximately 10 percent of the US population is gay. For the purposes of argument, let's make a more conservative estimate of 6%. Use this fact, along with some computations of probability, to argue that, in all likelihood, everyone knows someone who is gay.
2) A few years ago, one of my colleagues ended up with a Spanish class that was all boys-24 of them! When something like that happens, you tend to think "What are the chances of that?" You can figure that out. Assume the population of the school is half girls, half boys.
3) As you can see, the chance of getting an all-male class is extremely low. But you figured out the chance of a single occurrence. With all the classes given at ARHS over all the years, maybe that would happen by chance sooner or later. There are approximately 800 class sections formed over the course of a year. What is the chance of the all-boy class in problem #2 occurring at some point in a year? (Assume class size of 24 students and don't worry about taking into account that certain classes tend to attract one gender or another.)
4) Now think about ten years of problem #3. What are the chances that at least one all-boy class would happen at some point to somebody over the course of ten years?
5) This is part of the paradox of probability. On the one hand, when you see an unlikely event, you tend to think that it might be explained by something other than chance. For example, maybe the administrators deliberately gave that Spanish teacher all boys. On the other hand, over enough repetitions, if you wait long enough you may eventually see some of these unlikely events-and that too can be explained by chance. (Ha-ha, made you read something.)
6) At ARHS, approximately one fourth of the students are people of color. You walk into a classroom of 24 people, all white. Would it be reasonable to believe that this distribution occurred simply due to chance?
7) If the all-white class happened only once, you might explain it using the same reasoning as presented in #5. (Go back and read that if you didn't before.) Now think about the fact that the all-white class happens fairly frequently in our school. What difference does that frequency make in how you evaluate the situation?
8) Suppose a certain kind of cancer occurs at an average rate of 3 cases per hundred thousand people. In a town of 30,000 people, you find two cases of the disease. Is this an alarming statistic, or within the bounds of what might be expected? Compute the chances of no one in town getting the disease, then the chances of exactly one person getting it, then two people.
9) You are sitting on a jury in a medical malpractice case against an obstetrician. Complications during delivery caused the baby to suffer some brain damage. What type of statistical information would help you decide if the doctor is at fault or not?
10) Suppose a different disease occurs at a rate of 3 cases per million. Use this new rate to recompute question #8. (Again, you find two cases in a town of 30,000.) Is there a noticeable difference between the two situations?
11) How might you apply the reasoning in #5 to the situation in #10?
12) Increasing your understanding of chance helps you to evaluate things you see reported in the news. The media tends to prey on people's fears. Read the article about the autism cluster in New Jersey. Compute the probability that the cluster occurred due to chance, rather than due to some other factor. Make up your own mind about whether or not people in that town should be worried.
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