 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-2/17/c/iii/solution

P (B ∣ A) P (A)

equals

P (A \cap B)

, while

P (A ∣ B) = P (A \cap B) / P (B)

. Thus, wherever the original quotient is defined,

\frac{P ( B ∣ A ) P ( A )}{P ( A ∣ B )} = P (B) .

In particular a zero numerator and denominator cannot be cancelled; the original expression requires

P (A ∣ B) > 0

. This is the rearrangement of Bayes' theorem.

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