Each problem counts 20 points. You must show all your work and explain your reasoning to receive full credit.

1. Let S be a sample space and let A, B be events in S with . Compute under each of the following assumptions separately:
   1. if A and B are disjoint
   2. If A and B are independent
   3. If .
   
   

2. A fair die is rolled four times. A failure is defined as rolling a five or six, any other outcome is a success.
   1. Write down the probability of exactly k success for each k value, k=0,1,2,3, and 4.
   2. Simplify the probability of three successes and the probability of four successes from part a) to fractions in lowest terms and show that three is twice as likely as four.
   
   

3. An urn contains five balls, 2 red and three green. Two of the five are chosen randomly (without replacement) Find:
   1. The probability that both balls are the same color.
   2. The probability that at least one of the balls chosen is green.
   
   

4. Let X be a random variable which takes on the values 1,2,3, and 6 each with probability 1/4. Find:
   1. The expected value of X
   2. The expected value of X²
   3. The variance of X.
   
   

5. A two-headed coin is in an urn with four fair coins, making five coins in all. A coin is chosen randomly and tossed three times. Given that all three tosses yielded heads, what is the conditional probability that the two-headed coin was chosen?