 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ia/paper-2/10f/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

Express the three counts as sums of indicator variables for

B_{i} = 0

B_{i} = 1

and

B_{i} \geq 2

. The linearity of expectation applies even though the bin counts are dependent. The binomial distribution gives

p_{0} = (1 - 1/ m)^{n}, p_{1} = \frac{n}{m} (1 - 1/ m)^{n - 1} .

Thus

E [E] = m (1 - 1/ m)^{n}, E [S] = n (1 - 1/ m)^{n - 1},

E [C] = m [1 - (1 - 1/ m)^{n} - \frac{n}{m} (1 - 1/ m)^{n - 1}] .

The relation

E + S + C = m

checks the sum of these expectations. For

m = 1

, use the direct deterministic occupancies, or the usual binomial convention

0^{0} = 1

when

n = 1

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