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 :

Solution

Maths sir · 4 steps

Fix the repeated colour, then the odd ball's colour, then where the odd ball sits in the line of four.

1. Find p. With replacement, each draw has 6 equally likely colours, so two draws give 6² = 36 outcomes. Both balls match in 6 of them, so p = 6/36 = 1/6.
2. Count for q. Four draws give 6⁴ = 1296 outcomes. For exactly three of one colour, pick the repeated colour (6 ways). Pick a different colour for the fourth ball (5 ways). Pick which of the 4 draws holds the odd ball (4 ways).

   q = (6 × 5 × 4)/6⁴ = 120/1296 = 5/54

3. Take the ratio. p/q = 1/6 × 54/5 = 9/5. Since 9 and 5 are coprime, m = 9 and n = 5.
4. So. m + n = 9 + 5 = 14.

The common slip: 41. Leaving out the 4 positions of the odd ball gives q = 30/1296 and p:q = 36:5, so m + n = 41.

The idea to keep. Draws in succession are ordered outcomes. After you choose the colours, always multiply by the number of positions the odd one out can take.

Answer from the NTA final answer key. Solution by PrepWiser.