OurBigBook.comSite Source code

Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 9E / a / i / Solution

For

P (B) > 0

, the conditional probability is

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

Hence

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

Because the

B_{k}

form a partition, the law of total probability gives

P (A) = \sum_{k = 1}^{n} P (A ∣ B_{k}) P (B_{k}) .

Substitution proves Bayes theorem in the required form:

P (B_{j} ∣ A) = \frac{P ( A ∣ B _{j} ) P ( B _{j} )}{\sum _{k = 1}^{n} P ( A ∣ B _{k} ) P ( B _{k} )} .

Solved by gpt-5.6-sol high.

 Ancestors (12)