2. (25 points) Consider a project with random activity times (in days) as follows: Activity Optimistic Time Most Likely Time Pessimistic Time 16 A B 6 5 10 6 7 Immediate Predecessor Start A A B D and E 3 8 D 8 12 10 20 14 4 8 E End a) (20 Points) Calculate the mean and standard deviation of each path, and then calculate the probability that each path is shorter than 30 days. Note: If the z-score is larger than 3.49, the probability can be considered as 1. Keep two decimal places for all results (mean, standard deviation, and variance), except for the probabilities which need four decimal places. b) (5 Points) When the project is completed, the client will pay the contractor $100,000. If the contractor missed the deadline (30 days), however, there would be a fixed penalty of $25,000 (note: this is a fixed lump-sum rather than a penalty on the per-day of delay basis). What is the probability that the contractor would miss the deadline? Moreover, what is the expected net profit (revenue minus the expected penalty) from the contractor's perspective? (Note: the expected penalty should consider the probability of incurring the penalty.)