JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Probability

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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 colours is q. If

p : q = m : n

, where

m

and

n

are coprime, then

m + n

is equal to ____.

Answer: 14

Solution

With replacement, two draws share a colour with probability

p = 6 \cdot \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{6}

. For four draws with exactly three of one colour, the repeated colour can be chosen in

6

ways, its three positions in

(3 4) = 4

ways, and the odd ball's colour in

5

ways, so

q = \frac{6 \cdot 4 \cdot 5}{6 ^{4}} = \frac{5}{54}

. Then

p : q = \frac{1}{6} : \frac{5}{54} = 9 : 5

, and since

g cd (9, 5) = 1

m + n = 9 + 5 = 14

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Integer answer

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JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Probability

Solution

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