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 ( p / q ) where p and q are relatively prime positive integers. Find the value of (q-p).

Question

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 ( p / q ) where p and q are relatively prime positive integers. Find the value of (q-p). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/70de4d7e266252a8c667dccf084eb864-.png" /> S is the probability of success P((C/S))=(P(C) P((S/C))/P(A) P((S)A)+P(B) ⋅ P((S)B)+P(C) ⋅ P((S/C))=(0.7 × 0.4/0.1 × 0.8+0.2 × 0.6+0.7 × 0.4) =(7/12) q=12, p=7

S is the probability of success P(SC​)=P(A)P(AS​)+P(B)⋅P(BS​)+P(C)⋅P(CS​)P(C)P(CS​)​=0.1×0.8+0.2×0.6+0.7×0.40.7×0.4​ =127​ q=12,p=7

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