$1.$ The following table represents a plan for a project:

$\backslash$begin$\{$tabular$\}\{|$l$|$l$|$l$|$l$|$l$|\}$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Job $\backslash \backslash$

Num

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$l$\}$

Predecessor $\backslash \backslash$

Jobs

$\backslash$end$\{$tabular$\}$ & a & m & b $\backslash \backslash$

$\backslash$hline $1$ & & $2$ & $3$ & $4\backslash \backslash$

$\backslash$hline $2$ & $1$ & $1$ & $2$ & $3\backslash \backslash$

$\backslash$hline $3$ & $1$ & $4$ & $5$ & $12\backslash \backslash$

$\backslash$hline $4$ & $1$ & $3$ & $4$ & $11\backslash \backslash$

$\backslash$hline $5$ & $2$ & $1$ & $3$ & $5\backslash \backslash$

$\backslash$hline $6$ & $3$ & $1$ & $2$ & $3\backslash \backslash$

$\backslash$hline $7$ & $4$ & $1$ & $8$ & $9\backslash \backslash$

$\backslash$hline $8$ & $5,6$ & $2$ & $4$ & $6\backslash \backslash$

$\backslash$hline $9$ & $8$ & $2$ & $4$ & $12\backslash \backslash$

$\backslash$hline $10$ & $7$ & $3$ & $4$ & $5\backslash \backslash$

$\backslash$hline $11$ & $9,10$ & $5$ & $7$ & $8\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

a$.$ Construct the appropriate network diagram.

b$.$ Indicate the critical path.

c$.$ What is the expected completion time for the project?

d$.$ You can accomplish any one of the following at an additional cost of $\backslash (\backslash$$ $1,500\backslash )$ :

$1.$ Reduce job $5$ by two days.

$2.$ Reduce job $3$ by two days.

$3.$ Reduce job $7$ by two days.

If you save $\backslash (\backslash$$ $1,000\backslash )$ for each day that the earliest completion time is reduced, which action, if any, would you choose?

e$.$ What is the probability that the project will take more than $28$ days to complete?