Consider the following activities, their predecessors and the three time estimates $($days$)$:

Activity Predecessor Optimistic $($a$)$ Most Probable

$($m$)$ Pessimistic $($b$)$ Expected

Value Variance

A $246$

B A $357$

C B $135$

D B $81012$

E D $6810$

F D $345$

G D $468$

H C$,$E$,$F$,$G $789$

I H $357$

J H $369$

K I $369$

L J $123$

M K$,$L $61218$

$($a$)$ Compute the expected completion time the variance for each activity.

$($b$)$ Draw a network and determine the critical path based on the expected completion times.

$($c $)$ What is the probability of finishing the project in less than $48$ days?

$($d $)$ What is the probability of finishing the project in less than $51$ days?