Project 1. A bit of history. On page 409, we said, “The systematic study of randomness . . . began when seventeenth-century French gamblers asked French mathematicians for help in figuring out the ‘fair value’ of bets on games of chance.’’ Pierre de Fermat and Blaise Pascal were two of the mathematicians who responded. Both are interesting characters. Choose one of these men. Write a brief essay giving his dates, some anecdotes you find noteworthy from his life, and at least one example of a probability problem he studied. (A Web search on the name will produce abundant information. Remember to use your own words in writing your essay.)

Project 2. Reacting to risks. On page 418, we quoted a writer as saying, “Few of us would leave a baby sleeping alone in a house while we drove off on a 10-minute errand, even though car-crash risks are much greater than home risks.’’ Take it as a fact that the probability that the baby will be injured in the car is very much higher than the probability of any harm occurring at home in the same time period. Would you leave the baby alone? Explain your reasons in a short essay. If you would not leave the baby alone, be sure to explain why you choose to ignore the probabilities.

Project 3. First digits. Here is a remarkable fact: the first digits of the numbers in long tables are usually not equally likely to have any of the 10 possible values 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The digit 1 tends to occur with probability roughly 0.3, the digit 2 with probability about 0.17, and so on. You can find more information about this fact, called “Benford’s law,’’ on the Web or in two articles by Theodore P. Hill, “The Difficulty of Faking Data,’’ Chance, 12, No. 3 (1999), pp. 27–31; and “The First Digit Phenomenon,’’ American Scientist, 86 (1998), pp. 358–363. You don’t have to read these articles for this project.

Locate at least two long tables whose entries could plausibly begin with any digit. You may choose data tables, such as populations of many cities, the number of shares traded on the New York Stock Exchange on many days, or mathematical tables such as logarithms or square roots. We hope it’s clear that you can’t use the table of random digits. Let’s require that your examples each contain at least 300 numbers. Tally the first digits of all entries in each table. Report the distributions (in percentages) and compare them with each other, with Benford’s law, and with the “equally likely’’ distribution.

Project 4. Personal probability. Personal probabilities are personal, so we expect them to vary from person to person. Choose an event that most students at your school should have an opinion about, such as rain next Friday or a victory in your team’s next game. Ask many students (at least 50) to tell you what probability they would assign to rain or a victory. Then analyze the data with a graph and numbers—shape, center, spread, and all that. What do your data show about personal probabilities for this future event?

Project 5. Making decisions. Exercise 20.12 (page 476) reported the results of a study by the psychologist Amos Tversky on the effect of wording on people’s decisions about chance outcomes. His subjects were college students. Repeat Tversky’s study at your school. Prepare two typed cards. One says:

You are responsible for treating 600 people who have been exposed to a fatal virus. Treatment A has probability 1-in-2 of saving all 600 and probability 1-in-2 that all 600 will die. Treatment B is guaranteed to save exactly 400 of the 600 people. Which treatment will you give?

The second card says:

You are responsible for treating 600 people who have been exposed to a fatal virus. Treatment A has probability 1-in-2 of saving all 600 and probability 1-in-2 that all 600 will die. Treatment B will definitely lose exactly 200 of the lives. Which treatment will you give?

Show each card to at least 25 people (25 different people for each, chosen as randomly as you can conveniently manage and chosen from people who have not studied probability). Record the choices. Tversky claims that people shown the first card tend to choose B, while those shown the second card tend to choose A. Do your results agree with this claim? Write a brief summary of your findings: Do people use expected values in their decisions? Does the frame in which a decision is presented (the wording, for example) influence choices?