First Problem Set

Due Monday, September 17, 2007

1. At a certain racetrack, the payouts on a four-horse race are listed at 3:1, 4:1, 9:2 and 6:1 on horses A, B, C and D, respectively. For example, if you bet $2 on horse A, then you'll win $6 if it wins and nothing if it loses. Are these reasonable payouts for the racetrack to post? That is, is there a combination of bets that you can place on the four horses that will guarantee that you win, regardless of which horse wins the race? If there is such a combination, what bets would you place (multiples of $2 on each horse) that would guarantee your win? What would you expect to win if horse A, B, C or D wins?

2. Show that P(A&B)<=P(A)<=P(A or B). If P(A)=0.5 and P(B)=0.7, answer the following questions, explaining why your answers are correct:

a) Are A and B mutually exclusive? YES, NO, CANNOT TELL
b) Are A and B independent? YES, NO, CANNOT TELL
c) Is P(A or B)=1? YES, NO, CANNOT TELL
d) How would these answers change if you knew that P(A or B)=1? If you knew that P(A&B)=0.35?

3. Two fair dice are tossed. Define A=first die comes up even, B=second die comes up even, C=sum of two dies is even. Then

a) Are A and B independent? B and C? C and A? Explain.
b) Are A, B and C independent? Explain.

4. A joint probability distribution is given in the table:

Y

1

2

3

X

1

.03

.04

.05

2

.08

.06

.04

3

.16

.32

.22

Calculate: E(x); E(y); Var(x); Var(y); P(x>y); P(y|x=1); E(y|x=1); Var(y|x=1); P(w|w=x+y); E(w|w=x+y); Var(w|w=x+y)

5. Suppose that P(y=1 | x=1)=0.3, P(y=2 | x=1)=0.4 and P(x=1 & y=3)=0.05, that x takes on only values 1 and 2, and that y takes on only values 1, 2 and 3.

a) Assume that x and y are independent. Compute complete joint, conditional and marginal distributions that correspond to these figures (there may be more than one solution).
b) Give a second solution where x and y are not independent.

6. Suppose the incidence of breast cancer in the general population is 0.8%. If a woman has breast cancer, then 90% of the time it will result in a positive mammogram, whereas if a woman does not have the disease, a positive mammogram will result in 7% of the cases. Suppose a woman has a positive mammogram and no other risk factors.

a) What is the probability that the woman who tests positive has breast cancer?
b) Show how you would explain this to someone naive about probability using natural frequencies.
c) Draw a tree diagram for the problem and "flip the tree", demonstrating this way of solving these problems.

7. During the second world war, the allies were able to get a good estimate of the number of tanks that the Germans had produced by looking at the serial numbers of captured tanks and inferring the total number from the set of serial numbers. As it turned out, the Germans numbered their tanks sequentially from 1! (The allies, when they realized this, used a different numbering method that was not as susceptible to this kind of analysis). A simple version of the problem goes this way: Suppose you are in a new town and on your way to the hotel happen to notice the serial numbers of several taxis (including your own). Let's say you observe taxis numbered 37, 74 and 95. Assuming that taxis are numbered sequentially from 1, and choosing a reasonable prior (defend your choice of prior!), write down the posterior probability that the number of taxis in the city is N, as a function of N. Plot this for reasonable values of N. Try to come up with an estimate of the posterior mean, posterior variance, and posterior standard deviation.