Question

In a building programme the event that all the materials will be delivered at the correct time is $M$ and the event that the building programme will be completed on time is $F.$ Given that $P\left(M\right)=0.8$ and $P(M\cap F)=0.65$, find $P\left(F/M\right).$ If $P\left(F\right)=0.7$, find the probability that the building programme will be completed on time if all the materials are not delivered at the correct time.

• A. $P\left(F/M\right)=\frac{11}{16};P\left(F/\overline{M}\right)=\frac{1}{6}$
• B. $P\left(F/M\right)=\frac{15}{16};P\left(F/\overline{M}\right)=\frac{1}{8}$
• C. $P\left(F/M\right)=\frac{13}{16};P\left(F/\overline{M}\right)=\frac{1}{4}$
• D. None of these

Answer 1

Answer: (C)

$P\left(F/M\right)=\frac{13}{16};P\left(F/\overline{M}\right)=\frac{1}{4}$

Given,In building programme the event that all materials will be delivered at the correct time is $M$ and event that building programme will be completed on time is $F$.
$P\left(M\right)=0.8,P(M\cap F)=0.65$$P\left(F/M\right)=\frac{P(F\cap M)}{P\left(M\right)}=\frac{0.65}{0.8}=\frac{13}{16}$If $P\left(F\right)=0.7$
The probability that the building programme will be completed on tim if all materials are not delivered at the correct time=$P\left(F/\overline{M}\right)$
$\Rightarrow P\left(F/\overline{M}\right)=\frac{P(F\cap \overline{M})}{P\left(\overline{M}\right)}=\frac{\left(P\right(F)-P(F\cap M\left)\right)}{(1-P(M\left)\right)}$
$\Rightarrow P\left(F/\overline{M}\right)=\frac{(0.7-0.65)}{(1-0.8)}=\frac{0.05}{0.2}=\frac{1}{4}$