$\backslash$begin$\{$tabular$\}\{|$ccc$|\}\backslash$hline Task & Optimistic time & Pessimistic time $\backslash \backslash \backslash$hline $\backslash$hline A & $6$ & $20\backslash \backslash$ B & $7$ & $13\backslash \backslash$ C & $6$ & $14\backslash \backslash$ D & $5$ & $14\backslash \backslash \backslash$hline $\backslash$end$\{$tabular$\}3)(19$ pts$)$ You have been hired to plan and manage a small but important project for your company; this project consists of four tasks as indicated in the AON precedence network below:

The duration of all four tasks follow a uniform distribution, the optimistic and pessimistic times are indicated in the table below. You can assume that all four tasks are statistically independent.

a$)$ Using the Classic PERT model, what is the expected project makespan and the expected critical tasks? Using the expected task durations, what are the expected total slack values? Using the Classic PERT model, what is the probability that the project will be completed in $25$ weeks?

In the following questions, do NOT use the Classic PERT model:

b$)$ Using a Monte$-$Carlo simulation with $300$ trials, what is the expected makespan of the project? What is a $96$ percent confidence interval for the project makespan?

c$)$ If you want a $96$ percent chance that the project will be completed on time, when would you promise this project to your customer?