The following information has been compiled for a project that is about to begin. Assume the time estimates were made at the $99+$ percent level.

$\backslash$table$[[$Activity$,\backslash$table$[[$Preceding$],[$Activities$]],\backslash$table$[[$Optimistic$],[$Time$]],\backslash$table$[[$Most$],[$Likely$]],\backslash$table$[[$Pessimistic$],[$Time$]]],[$A$,$None,$6,7,14],[$B$,$None,$8,10,12],[$C$,$A$,2,3,4],[$D$,$A$,6,7,8],[$E$,$B$,$ C$,5,5\mathrm{.}5,9],[$F$,$B$,$ C$,5,7,9],[$G$,$D$,$ E$,4,6,8],[$H$,$F$,2\mathrm{.}5,3,3\mathrm{.}5]]$

a$)$ Construct the network $(10$ points$).$

b$)$ What is the probability that the path with the longest expected duration will be completed $21$ days? $(10$ points$)$

c$)$ Assuming the paths are independent, what is the probability that the project will be comple within $21$ days? $(5$ points$)$