Background Info: Managers can belong to one of two groups: Group Y or Group Z. Group membership is known to everyone. Managers can be either “good” or “bad.” No one, even the managers, themselves, knows whether a particular manager is good or bad. However, group membership, does provide some information. It is commonly believed that 3/4 of the managers in group Y are good and that 1/4 of the managers in group Z are good. Firms hire managers to invest in projects. Good managers have a 2/3 probability of success when investing in a project; bad managers have a 1/2 probability of success when investing in the project.
Problem: Suppose managers from both groups repeatedly invest in projects. A specific manager’s project outcomes are independent of one another and project outcomes for different managers are independent of one another as well. Suppose a manager from group Y continually invests and fails. Suppose a manager from group Z continually invests and succeeds. How many failures from a Y member and successes from a Z member will it take before the manager from group Z is believed to be good with a higher probability than the manager from group Y?
I did some figuring and the probability that a manager from group Y is good and fails is 2/3. The probability that a manager from group Z is good and succeeds is 4/13. I'm very confused about this problem, any help would be greatly appreciated.