Question: A company with n women and m men as employees is deciding which employees to promote.
(a) Suppose for this part that the company decides to promote t employees, where 1 ≤ t ≤ n + m, by choosing t random employees (with equal probabilities for each set of t employees). What is the distribution of the number of women who get promoted?
(b) 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. Find the distributions of the number of women who are promoted, the number of women who are not promoted, and the number of employees who are promoted.
(c) In the set-up from (b), find the conditional distribution of the number of women who are promoted, given that exactly t employees are promoted.