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 manager 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) Show that the expected number of projects for which you are a project manager is 7/3. Hint: 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. Use the same logic as part (c) You could also solve this problem using linearity of expectation e) Show that the variance in the number of projects for which you are a manager is 14/9. Hint: along with your answer to part (2d), compute E(X2) using the definition of expected value in order to calculate V(x) E(X2)- E(X)2 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 manager 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) Show that the expected number of projects for which you are a project manager is 7/3. Hint: 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. Use the same logic as part (c) You could also solve this problem using linearity of expectation e) Show that the variance in the number of projects for which you are a manager is 14/9. Hint: along with your answer to part (2d), compute E(X2) using the definition of expected value in order to calculate V(x) E(X2)- E(X)2