A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is Next four balls are drawn in succession with replacement and the probability that exactly thr

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Solution

Q.
A bag contains six balls of different colours. Two balls are drawn in succession with replacement. The probability that both the balls are of the same colour is $p .$ Next four balls are drawn in succession with replacement and the probability that exactly three balls are of the same colour is $q$ . If $p : q = m : n$ ,where $m$ and $n$ are coprime, then $m + n$ is equal to _______. [2023]

Ans.
(14)

Bag contains six balls of different colours.

Probability of drawing a ball of one colour $= \frac{1}{6}$

$∴$ $Probability of drawing two balls of same colour with replacement = p = {(\frac{1}{6})}^{2}$

Probability that exactly three balls are of same colour when four balls are drawn in succession with replacement.

$= q = 4 {(\frac{1}{6})}^{3} \times \frac{5}{6} = \frac{20}{6^{4}}$

$p : q = \frac{1}{6^{2}} : \frac{20}{6^{4}} = \frac{36}{20} = \frac{9}{5}$

$p : q = m : n = 9 : 5$ $∴$ $m = 9, n = 5$

$Now, m + n = 9 + 5 = 14$

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