 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/16/a/i/solution

With

N /2

p_{w} = \frac{N /2}{N} \frac{N /2 - 1}{N - 1} = \frac{N - 2}{4 ( N - 1 )} .

Rearranging,

4 p_{w} (N - 1) = N - 2

, so

N = \frac{2 - 4 p _{w}}{1 - 4 p _{w}} = \frac{2 ( 1 - 2 p _{w} )}{1 - 4 p _{w}} .

A finite admissible population has even

N \geq 2

and

0 \leq p_{w} < 1/4

, so the denominator is nonzero.

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