A construction project is about $2$ months away from the

scheduled completion date. Based on the progress of the project

to date, the contractor estimated that the project can be completed

on schedule without difficulty if good weather continues for the

next $2$ months; whereas if the weather in the next $2$ months is

normal, the probability of on$-$time completion will be $90\%.$ This

probability will be reduced to $20\%$ if bad weather prevails, in

which case he has the option of launching a crash program to

improve the on$-$time completion probability to $80\%.$ However,

because of uncertainty in the labor market, there is only a $50$

$50$ chance that he can successfully launch a crash program when

needed. Assume that no crash program is necessary if the weather

condition is good or normal. Suppose also that the weather bu$-$

reau predicts that the relative likelihoods of good, normal, and

bad weather conditions in the next $2$ months are $1$:$2$:$2.$

$($a$)$ What is the probability that the construction project will be

completed on schedule?

$($b$)$ If the project was not completed on schedule, what is the

probability that the weather condition had been normal?