Problem $1.(20$ pints$)$

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

$\backslash$table$[[$Activity$,$Predecessor,Optimistic $($a$),$Most Probable $($m$),$Pessimistic $($b$),\backslash$table$[[$Expected$],[$Value$]],$Variance$],[$A$,,2,4,6,,],[$B$,$A$,3,5,7,,],[$C$,$B$,1,3,5,,],[$D$,$B$,8,10,12,,],[$E$,$D$,6,8,10,,],[$F$,$D$,3,4,5,,],[$G$,$D$,4,6,8,,],[$H$,$C$,$E$,$F$,$G$,7,8,9,,],[$I$,$H$,3,5,7,,],[$J$,$H$,3,6,9,,],[$G$,$I,$3,6,9,,],[7,$J$,1,2,3,,],[$L$,$K$,$L$,6,12,18,,],[$M$,,,,,,],[,,,,,,]]$

$($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?