The table below lists three time estimates and the immediate predecessor$($s$)$ for each activity in a construction project that is to be scheduled using PERT. The three time estimates $($in weeks$)$ are in the order: optimistic time, most likely time, and pessimistic time.

Activity

$1$

$2$

$3$

$4$

$5$

$6$

$7$

$8$

Time estimates $($weeks$)$

$1,2,3$

$3,5,7$

$5,7,9$

$5,7,12$

$1,2,3$

$2,3,4$

$7,9,11$

$1,3,8$

Immediate predecessor

$-$

$1$

$1$

$2$

$3$

$4,5$

$3$

$6,7$

What is the probability that the project will be completed within $20$ weeks?

$45\%$

$86\%$

$31\%$

$16\%$