Each problem counts 20 points except for problem 3 as noted.

1. A fair die is rolled once. Let X = the number obtained on the die, and let Y = 1/X.
   1. Write down the probability density function for each of X and Y.
   2. Compute the expected value of each of X and Y. (Fraction, no decimals, but you need not do the arithmetic.)
   3. Which is larger, E(Y) or 1/(E(X)) ?
   
   

2. The sample space S contains events E and F. Pr(E) = 7/16; Pr(F) = 3/8, and Pr(E |F) = 1/6. Give the following answers as fractions in lowest terms.
   1. Compute Pr(E F).
   2. Compute Pr(E F).
   3. Compute Pr(F|E ).
   4. Compute Pr( E F | E F).
   
   

3. An urn contains two fair coins and one two-headed coin. A coin is chosen randomly and tossed.
   1. What is the probability that the toss results in a head?

   2. 10 quiz points

      If the coin is tossed a second time, find the conditional probability that the second toss is heads given that the first one was.
   
   

4. A club has fourteen members. Six of the members are M.D.'s and four are professional golfers. (No one is both!) A meeting-site committee of three is chosen randomly every month.
   1. Find the probability that the January 2001 committee is all M.D.'s.
   2. Find the probability that the March 2003 committee has no M.D.'s and no pro golfers on it.
   3. Find the probability that at most one committee member for May 2000 is neither an M.D. nor a pro golfer.
   
   

5. A sample space S contains two events, A and B with Pr(A)=1/3 and Pr(B) = 1/4. Compute Pr(A B) and Pr(A B) under each of the following scenarios, or side conditions.
   1. B A
   2. A and B are disjoint.
   3. A and B are independent.
   4. Pr(A|B) = 1/8.