← Mathematics — Year 1/Probability and Statistics I: Models and Inference

Bayes' Theorem

≈ 25 min

0/4 practice checks

Bayes' Theorem

Bayes' theorem reverses conditioning by combining a likelihood with a prior and normalising by total evidence. $P(A\mid B)=P(B\mid A)P(A)/P(B)$ when $P(B)>0$ . The distinction between a formal hypothesis and an intuitive picture is made explicit so that calculations can be justified, not merely patterned.

Worked reasoning

Apply Bayes: $0.5(0.4)/0.25$ .
$0.2/0.25=0.8$ .

Key idea

$P(A\mid B)=P(B\mid A)P(A)/P(B)$ when $P(B)>0$ .

If $P(A)=0.4$ , $P(B|A)=0.5$ and $P(B)=0.25$ , find $P(A|B)$ .

Which statement best captures the central mathematical idea in Bayes' Theorem?

$P(A\mid B)=P(B\mid A)P(A)/P(B)$ when $P(B)>0$ .

A familiar formula remains valid after its hypotheses, domain or convergence conditions are removed.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Bayes' Theorem, which move is most reliable?

Write the governing definition and list its hypotheses before calculating in bayes' theorem.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Bayes' Theorem?

A familiar formula remains valid after its hypotheses, domain or convergence conditions are removed.

$P(A\mid B)=P(B\mid A)P(A)/P(B)$ when $P(B)>0$ .

Write the governing definition and list its hypotheses before calculating in bayes' theorem.

Test a representative example and a boundary or counterexample, then verify the conclusion directly from the definition.

PreviousBernoulli and Binomial Distributions NextSampling Variability and Confidence Intervals