Each problem counts 20 points, except that parts e) and f) of problem # 3 count 5 bonus quiz points each.

1. Define explicitly two sets A and B satisfying n(A)=7, n(B)=9, and n(A B)=12.
   1. How many elements are in
   2. How many elements are in

2. Let E and F be events in a sample space S. Suppose P(E)= 2/5, P(F)= 3/5, and P(E F) = 4/5. Find P(E F) and P(E' F)
   

3. A four digit number is formed only using nonzero digits.
   1. How many different numbers are possible?
   2. How many are possible if no digit can be used more than once?
   3. How many are possible if no digit can be used more than once and the digits used are all odd?
   4. How many are possible if each digit used must occur exactly twice?
   5. (BONUS) How many are possible if each digit used must appear more than once?
   6. (BONUS) How many are possible if the four digits must be all different and and must appear in decreasing order (for example, 8 4 3 1)?
   
   

4. Eight men and six women are members of a club. The club chooses its officers randomly rather than by election. Three officers - a chair, secretary, and treasurer - are to be chosen. Define a sample space for this experiment and find the probability that:
   1. All officeres chosen are the same sex.
   2. The secretary and treasurer are not the same sex.
   
   

5. In a certain sport a championship series is played as follows: The challenger, A, must win three games to win the championship, while the champion B, needs to win only two games to retain the championship. Games are played over and over until either A has won three games or B, two games. Ties are not possible.
   1. How many sequences of winners (A's, B's) are possible?
   2. What is the largest possible number of games which could be played?