Bayes' Theorem

Reverse a conditional probability: go from P(evidence | cause) to P(cause | evidence).

What is it?

Bayes' theorem expresses $P (A ∣ B)$ in terms of $P (B ∣ A)$ , the prior $P (A)$ and the total probability of the evidence $P (B)$ .

Why is it useful?

Interviewers love questions where the intuitive answer ignores the base rate. Bayes forces you to weigh how likely the evidence is under every hypothesis.

When should you think of it?

Reach for it when you are told how likely an observation is given a hidden state, and asked how likely the hidden state is given the observation.

Think of this method when…

The question gives $P (observation ∣ state)$ but asks for the reverse
There is a hidden type: fair vs biased coin, sick vs healthy, informed vs noise trader
Wording like "given that the test is positive" or "you observe… what is the probability that…"
A rare base rate combined with an imperfect signal
Sequential evidence that should update a belief

Core idea

Posterior is proportional to prior times likelihood. Normalize by summing prior × likelihood over every hypothesis.

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{\sum _{i} P ( B ∣ A _{i} ) P ( A _{i} )}

Step-by-step playbook

1List the hypotheses $A_{1}, \dots, A_{k}$ and write their priors.
2Write the likelihood of the observed evidence under each hypothesis.
3Multiply prior by likelihood for each hypothesis.
4Divide the target product by the sum of all products.
5Sanity-check: a rare prior should pull the posterior down.

Worked example

A disease affects $1%$ of people. A test detects it $99%$ of the time and gives a false positive $5%$ of the time. A person tests positive. What is the probability they have the disease?

1
Prior × likelihood for "sick": $0.01 \times 0.99 = 0.0099$ .
2
Prior × likelihood for "healthy": $0.99 \times 0.05 = 0.0495$ .
3
Normalize.
$P (sick ∣ +) = \frac{0.0099}{0.0099 + 0.0495} = \frac{0.0099}{0.0594} = \frac{1}{6}$
Answer: $1/6 \approx 16.7%$

Common mistakes

Confusing $P (A ∣ B)$ with $P (B ∣ A)$ (the prosecutor's fallacy).
Forgetting the false-positive branch in the denominator.
Ignoring the base rate because the test sounds accurate.

Would you use this method?

One of two coins is double-headed. You pick one at random and flip heads three times. How likely is it the double-headed coin?

Practice this method

33 QuantSolve questions are tagged with this method.

Practice this method

Related lessons & methods

Bayes' Theorem · Probability Conditional Probability · Probability Bayesian Updating with Conjugate Priors · Statistics

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