Box $A$ contains 2 black and 3 red balls. While box $B$ contains 3 black and 4 red balls. Out of these two boxes one is selected at random; and the probability of choosing box $A$ is double that of box $B$. If a red ball is drawn from the selected box, then the probability that it has come from box$B$. is

Question

Dr. Abubacker · Accepted Answer

(b)
Let probability of box $B,\ P(B) = P$
According to given condition
$P(A) = 2P(B) = 2P$
Now, $P\left( \frac{R}{A} ight) = \frac{\ ^{3}C_{1}}{\ ^{5}C_{1}} = \frac{3}{5}$
and $P\left( \frac{R}{B} ight) = \frac{\ ^{4}C_{1}}{\ ^{7}C_{1}} = \frac{4}{7}$
$	herefore\ P\left( \frac{B}{R} ight) = \frac{P(B).P\left( \frac{R}{B} ight)}{P(A).P\left( \frac{R}{A} ight) + P(B).P\left( \frac{R}{B} ight)}$
$= \frac{p.\frac{4}{7}}{2p.\frac{3}{5} + p.\frac{4}{7}} = \frac{10}{31}$

Explanation

Reviewed by Dr. Abubacker

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