The table below lists $14$ activities involved in a project by a consulting company hired to audit and

evaluate a manufacturer$$s logistics operations. The critical path is A$-$C$-$E$-$H$-$I$-$K$-$M$-$N$.$

Activity Immediate

Predecessors

Optimistic $($a$)$

$($weeks$)$

Most Likely $($m$)$

$($weeks$)$

Pessimistic $($b$)$

$($weeks$)$

A None $4\mathrm{.}06\mathrm{.}08\mathrm{.}0$

B None $1\mathrm{.}02\mathrm{.}03\mathrm{.}0$

C A $6\mathrm{.}06\mathrm{.}06\mathrm{.}0$

D A $6\mathrm{.}08\mathrm{.}010\mathrm{.}0$

E B$,$ C $1\mathrm{.}09\mathrm{.}014\mathrm{.}0$

F D $2\mathrm{.}03\mathrm{.}07\mathrm{.}0$

G D $1\mathrm{.}07\mathrm{.}010\mathrm{.}0$

H E$,$ F $4\mathrm{.}05\mathrm{.}06\mathrm{.}0$

I G$,$ H $1\mathrm{.}06\mathrm{.}08\mathrm{.}0$

J I $2\mathrm{.}05\mathrm{.}011\mathrm{.}0$

K I $8\mathrm{.}09\mathrm{.}013\mathrm{.}0$

L J $2\mathrm{.}04\mathrm{.}06\mathrm{.}0$

M K $1\mathrm{.}02\mathrm{.}03\mathrm{.}0$

N L$,$ M $6\mathrm{.}08\mathrm{.}010\mathrm{.}0$

$5$a$)$ What is the standard deviation of the project?

$5$b$)$ What is the critical path?

$5$c$)$ What is the expected project duration?

$5$d$)$ What is the probability that the project can be completed within $45$ weeks?

$5$e$)$ What is the probability that it will take more than $55$ weeks to complete the project?

$5$f$)$ What is the probability that the project will be completed in between $45$ and $55$ weeks?