 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-2/18/c/solution

Condition on whether the first selection was a red apple. Its probability is

3/18

. If it was, two red apples remain among seventeen fruits; otherwise three remain. The law of total probability gives

P (second is red apple) = \frac{3}{18} \frac{2}{17} + \frac{15}{18} \frac{3}{17} = \frac{1}{6} .

The same result follows from exchangeability: before either selection is observed, each ordered draw position has the original red-apple fraction

3/18

, despite dependence between the draws.

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