A partially solved PERT problem is detailed in the table below. Times are given in weeks.

$\backslash$table$[[$Activity$,$Preceding,$\backslash$table$[[$Optimistic$],[$Time$]],\backslash$table$[[$Probable$],[$Time$]],\backslash$table$[[$Pessimistic$],[$Time$]],\backslash$table$[[$Expected$],[$Time$]],$Variance$],[-,7,10,19,,],[$B$,$A$,2,2,8,,],[$C$,$A$,8,12,16,,],[$D$,$A$,3,5,10,,],[$E$,$B$,4,6,8,,],[$F$,$B$,6,8,10,,],[$G$,$C$,$ F$,2,3,4,,],[$H$,$D$,2,2,8,,],[$I$,$H$,6,8,16,,],[$J$,$G$,$ I,$4,6,14,,],[$K$,$E$,$ J$,2,5,8,,]]$

$($a$)$ Calculate the expected time and variance for each activity. Enter these values in the appropriate columns in the table above.

$($b$)$ Which activities form the critical path?

$($c$)$ What is the estimated time of the critical path?

$($d$)$ What are the project variance and the project standard deviation?

$($e$)$ What is the probability of completion of the project before week $42?$