Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 10

Table of contents
- a 10
  - Solution a
- b 10
  - Solution b

a

Solution

Let

S

denote singing, and

M, F

denote male and female. Equal population counts give

P (M) = P (F) = 1/2

. The law of total probability therefore yields

P (S) = pP (M) + qP (F) = \frac{p + q}{2} .

b

Solution

By Bayes' theorem, provided

p + q > 0

,

P (M ∣ S) = \frac{P ( S ∣ M ) P ( M )}{P ( S )} = \frac{p}{p + q} .

If

p = q = 0

, singing has probability zero and the stated observation cannot occur in this model; that conditional probability is undefined.

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