$(20$ POINTS$)$ The table below lists the activity times and precedence requirements for the

activities required to complete a new equipment installation project in a factory. Three time

estimates $($in weeks$)$ are given for each activity: optimistic time, most likely time, and pessimistic

time. $($a$)$ Draw the PERT network diagram for the project, labeling all activities and times. $($b$)$

Perform a forward pass and backward pass through the network to determine the critical path,

the expected duration of the project, and the slack time for each activity. $($c$)$ What is the

probability that the project will be completed one week or more before the expected

completion time?