Question:
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 :
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 :
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\(p = 6\cdot\left(\frac{1}{6}\right)^2\). For \(q\), choose the colour (6 ways), the 3 positions (4 ways) and a different colour for the remaining ball (5 ways) out of \(6^4\) outcomes.
Updated On: Nov 2, 2026
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Correct Answer: 14
Solution and Explanation
Two draws with replacement: the second ball must match the first, so \(p = \frac{6}{36} = \frac{1}{6}\).
Four draws with replacement, exactly three of the same colour: choose that colour (6 ways), choose which 3 of the 4 draws have it (\(\binom{4}{3} = 4\) ways), and the remaining ball must be one of the other 5 colours (5 ways).
Favourable outcomes \(= 6\cdot 4\cdot 5 = 120\); total \(= 6^4 = 1296\). So \(q = \frac{120}{1296} = \frac{5}{54}\).
\(p : q = \frac{1}{6} : \frac{5}{54} = 9 : 5\), with 9 and 5 coprime. So \(m = 9, n = 5\) and \(m + n = 14\).
Answer: 14
Four draws with replacement, exactly three of the same colour: choose that colour (6 ways), choose which 3 of the 4 draws have it (\(\binom{4}{3} = 4\) ways), and the remaining ball must be one of the other 5 colours (5 ways).
Favourable outcomes \(= 6\cdot 4\cdot 5 = 120\); total \(= 6^4 = 1296\). So \(q = \frac{120}{1296} = \frac{5}{54}\).
\(p : q = \frac{1}{6} : \frac{5}{54} = 9 : 5\), with 9 and 5 coprime. So \(m = 9, n = 5\) and \(m + n = 14\).
Answer: 14
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