 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2020/paper-1/16/b/ii/solution

Write

N = \sum_{j = 1}^{10} X_{j}

, where the independent Bernoulli random variables indicate rottenness. Since

X_{j}^{2} = X_{j}

, their variances are

E [X_{j}^{2}] - E [X_{j}]^{2} = p - p^{2}

. The independence makes all cross covariances zero, so

Var (N) = 10 p (1 - p), σ_{N} = 10 p (1 - p) pieces .

p = 0

and

p = 1

, when the count is certain.

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