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Q. AJEE 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 believe that he will study $10 hr$., $7 hr$. and $4 hr /$ 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 positive integers. Find the value of $(q-p)$.

Probability - Part 2

Solution:

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$S$ is the probability of success
$P\left(\frac{C}{S}\right)=\frac{P(C) P\left(\frac{S}{C}\right)}{P(A) P\left(\frac{S}{A}\right)+P(B) \cdot P\left(\frac{S}{B}\right)+P(C) \cdot P\left(\frac{S}{C}\right)}=\frac{0.7 \times 0.4}{0.1 \times 0.8+0.2 \times 0.6+0.7 \times 0.4}$
$=\frac{7}{12} $
$q=12, p=7 $