 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/b/ii/solution

The probability of two black balls is the complement of the other two possibilities:

P (BB) = 1 - p_{w} - p_{d} = 1 - 2 x + p_{w} .

The probability of equal colours is

P (WW) + P (BB) = 1 - 2 x + 2 p_{w}

P (BB ∣ same colour) = \frac{1 - 2 x + p _{w}}{1 - 2 x + 2 p _{w}} .

This is defined when the conditioning event has positive probability. If

N = 2

with one ball of each colour, that event is impossible and the conditional probability is undefined.

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