Question

A JEE aspirant estimates that he will be successful with an $80\%$ chance if he studies $10$ hr/day, with $60\%$ chance if he studies $7$ hr/day, and with $40\%$ chance if he studies $4$ hr/day. Further, he believes that he will study $10$ hr, $7$ hr and $4$ hr per day with probability $0.1,$ $0.2$ and $0.7$ respectively. Given that he is successful, the probability that he studies for $4$ hr/day equals $\frac{p}{q}$ where $p$ and $q$ are relatively prime. Then the value of $(q-p)$ is

Answer 1

Let $S$ be the event that aspirant is successful.
Then, $P\left(C|S\right)=\frac{P\left(C\right)\cdot P\left(S|C\right)}{P\left(A\right)\cdot P\left(S|A\right)+P\left(B\right)\cdot P\left(S|B\right)+P\left(C\right)\cdot P\left(S|C\right)}$
$\Rightarrow P\left(C|S\right)=\frac{0.7\times 0.4}{0.1\times 0.8+0.2\times 0.6+0.7\times 0.4}$
$\Rightarrow P\left(C|S\right)=\frac{7}{12}=\frac{p}{q}$
$\therefore p=7,q=12$
Hence the value of $q-p$ is $5.$