1. A hat contains four slips of paper, numbered 1, 2, 3, and 4. Two slips are drawn at random without replacement. X is the number on the first slip and Y the sum of the two numbers drawn. Write in tabular form the joint pmf of .X; Y /. Hence find the marginal pmfs. Are X and Y independent?
2. (Conditional Expectation in Bridge). Let X be the number of clubs in the hand of North and Y the number of clubs in the hand of South in a bridge game. Write a general formula for E.X jY D y/, and compute E.X jY D 3/. Can you compute E.Y jX D 3/?
3. A fair die is rolled four times. Find the probabilities that
(a) at least one six is obtained;
(b) exactly one six and exactly one two is obtained;
(c) exactly one six, one two, and two fours are obtained.