At a local university, the Student Commission on Programming and Entertainment $($SCOPE$)$ is preparing to host its first rock concert of the school year. To successfully produce this rock concert, SCOPE has listed the requisite activities and related information in the following table $($duration estimates measured in days$).$

$\backslash$table$[[$Activity$,\backslash$table$[[$Immediate$],[$Predecessor$($s$)]],\backslash$table$[[$Optimistic$]],\backslash$table$[[$Most$],[$Probable$]],$Pessimistic$],[$A: Negotiate contract with,$-,8,10,15],[\backslash$table$[[$selected musicians$]],-,7,8,9],[\backslash$table$[[$B: Reserve site$],[$C: Manage travel logistics$],[$for music group$]],$A$,5,6,10],[$D: Screen & hire,,,,$],[,$ security personnel,B$,3,3,3],[$E: Arrange advertising,B$,$ C$,1,5,9],[,$ & ticketing,,,$7,10],[$F: Hire parking staff,D$,4,8,10]]$

a$.$ Draw the project network.

b$.$ Compute the expected duration and variance of each activity.

c$.$ Determine the critical path in the project network.

d$.$ What is the expected duration and variance of the critical path?

e$.$ Based only on the critical path, what is the likelihood that the project will be completed within $30$ days?

f$.$ If activity B is delayed by six days beyond its early start time, how does this affect the expected project duration?

g$.$ Using all paths through the project network, estimate the probability that the project will be completed within $30$ days. Compare your answer to the answer in part e and explain.