Please do all parts!

2. Suppose you are the most recently hired employee at a small startup with 9 employees, including you. For each project, 3 employees are randomly assigned to be project managers, and the remaining 6 employees work as software engineers. Suppose the company works on 7 different projects, which we will call Project 1 through Project 7. (a) For Project 1, before the roles are assigned, what is the probability that the single person who was first hired becomes a project manager? You meet privately with your boss and you learn that for Project 1, you will be a software engineer, not a project manager. After learning this, what is the probability that the person who was first hired is a project manager? How can knowing only your own role affect the chance of someone else becoming a project manager? (b) What is the probability that the person hired immediately before you is a project mangager for Projects 6 and 7, if he or she was a project manager for Projects 1 through 5? (c) What is the probability that you are a project manager for exactly 2 of the 7 projects? (d) What is the expected number of projects for which you are a project manager Solution: To use the definition of expected value, we need to know the probability that you manage k projects, where k can be any number from 0 through 7. Using the same logic as part (c), the probability you manage k projects is Therefore, using the definition of expected value, the expected number of times you are a project manager is