 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/16/b/i/solution

The probability that the first ball is white is

x

. Partitioning that event according to the second colour shows that

P (W B) = x - p_{w}

. Uniform sampling without replacement makes the two draw positions exchangeable, so

P (B W) = P (W B)

. Therefore

p_{d} = 2 (x - p_{w}) .

For a direct count, the same value is

2 (x N) ((1 - x) N) / [N (N - 1)]

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