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 th ree balls are of the same colour is . If , where and are coprime, then is equal to ________.

14 Explanation: Solution 1. Find the value of Two balls are drawn with replacement. For both balls to be the same color, the first ball can be any of the 6 colors, and the second ball must match it. Total outcomes = . Favorable outcomes = 6 (e.g., Red-Red, Blue-Blue, etc.). 2. Find the value of Four balls are drawn with replacement. We need exactly three balls to be of the same color. Step A: Choose which of the 6 colors will appear three times: ways. Step B: Choose which 3 positions out of the 4 draws will have that color: ways. Step C: The remaining 1 position must be any of the other 5 colors: ways. Total outcomes: . 3. Find the ratio Here, and . 4. Calculate Since 9 and 5 are coprime (their only common factor is 1): Final Answer:

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

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 ________.

JEE 2023 — Mathematics PYQ

Choose the correct answer:

Explanation

Solution

1. Find the value of $p$

2. Find the value of $q$

3. Find the ratio $p : q$

4. Calculate $m + n$