1. There are 10 slips of paper in a hat, each of which has one of the numbers 1, 2, . . . , 10 written on it, with no number appearing more than once. Two of the slips are randomly drawn, one at a time "without replacement" (with all sets of two slips equally likely to be chosen).

(a) (2 pts) What is the distribution of the number of drawn slips that have a value of at least 8 written on them?

(b) (2 pts) What is the distribution of the value of the j-th draw (for 1 ? j ? 2)?

3. A certain company has n + m employees, consisting of n women and m men. The company is deciding which employees to promote. (a) (3 pts) Suppose for this part that the company decides to promote 15 employees, where 1 g t S n + m, by choosing at random employees (with equal probabilities for each set of t employees). What is the distribution of an r.v. X that represents the number of women who get promoted? Now suppose that instead of having a predetermined number of promotions to give, the company decides independently for each employee, promoting the employee with probability p. (b) (3 pts) Find the distributions of the number of women who are promoted. (c) (3 pts) Find the distributions of the number of women who are not promoted. (d) (3 pts) Find the distributions of the number of employees who are promoted. (e) (3 pts) Find the conditional distribution of the number of women who are promoted, given that exactly t employees are promoted