 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/4/a/iv/solution

W > k

occurs exactly when both

X_{1} > k

and

X_{2} > k

P (W > k) = q^{k} q^{k} = q^{2 k} = (1 - (2 p - p^{2}))^{k} .

Thus

W \sim Geom (2 p - p^{2}) .

The new success probability

2 p - p^{2}

is the chance that at least one of the two trials succeeds.

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