We discussed the first quiz. Here are my comments, question by question:

1. People had two approaches to this problem. Some tried to estimate the number of faculty and their salaries, and multiply. It turns out that UVM has about 1300 full and part time faculty; most estimates were low. Also, when using this method, you need to include supporting people like secretaries, TAs and so forth that are also in the teaching budget. The other group estimated the tuition paid per student and the number of students. A fraction, perhaps half of this, is devoted to the teaching budget. The estimates using the first method were low for the reasons I mentioned, but if they are fixed, the two numbers are pretty consistent.

2. On this problem, the biggest difficulty was realizing that the likelihood for SON=Door 1 is of course 1/3 (most if not all got that right), because Monty has three doors to chose from. For SON=Door 2 it is of course 0. The problem arose with doors 3 and 4. For these doors, the likelihood is 1/2, because Monty only has two doors to choose from. For example, if SON=Door 3, Monty can’t open Door 3 (he always shows a goat), and he can’t open Door 1 (you chose that one), so he must choose either Door 2 or Door 4. There’s a 50% chance he’ll choose Door 2

3. This one is sampling with replacement, so the probability of observing each cab doesn’t change from cab to cab. For example, after you’ve been in Cab #3, and been let off at your hotel, the cab continues to drive around and can be seen again (and in fact you saw it again the next day.) This means that the likelihood for N cabs is 1/N^{3} for N≥4 and 0 for N<4.

4. Except for minor errors everyone got the first part of this right. However some thought that you should use the posterior probability of no drugs|positive test, and multiply it by the number of students. That’s not right. The correct way to do it is to take all the non-drug taking students (1470) and multiply by the probability of a false positive in a student who hasn’t taken the drugs (0.03 or 3%). That gives an expected number of students falsely testing positive at about 44.

5. The table is dependent, because if you compute the marginals across and down, and multiply pairwise, you don’t get the entry in the table. There is therefore precisely one independent table. You get it by multiplying the marginals pairwise and entering the products in the corresponding boxes in the table.

6. Here is the decision tree we drew.

The decision diagram for the quiz:

We then did the exercise of estimating each student’s personal utility function. Here is the result, for five of the students.

Several things are clear here. First, people are very risk-averse. Only one student thought that getting nothing for sure was about as good as an even bet on either winning $100,000 or losing as much as $25,000. Most people were only willing to risk losing a few thousand. The upper part of the curve also shows that people do not value the second $25,000 as much as the first, the third $25,000 as much as the second (with the exception of one of the red curves which shows a “kink”), and the fourth $25,000 as much as the third. The “kink” may just be an artifact; I don’t take it seriously.

Taken to someone like Bill Gates, who has an enormous amount of money, the marginal value of a few billion dollars is virtually nil to him. Even if there were not tax deductions for charitable giving, Bill and Melinda Gates are probably getting a lot more pleasure by giving this money away to help solve societal problems (e.g., eliminating polio and malaria) than they could ever get out of spending that money on themselves.

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