 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-209/2/a2/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Write

M = # H_{N}

X = # C_{N}

, and

q = P (X \geq θM /2)

. Since

0 \leq X \leq M

θM \leq E X \leq \frac{θM}{2} (1 - q) + Mq .

Thus

q \geq (θ /2) / (1 - θ /2) \geq θ /2

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