Bayes' Theorem Formula

Bayes' theorem gives the posterior probability of a hypothesis given evidence: P(H|E) = P(E|H) x P(H)/P(E).

The Formula

P(A∣B)=P(B∣A)⋅P(A)P(B)P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}

When to use: Start with a prior belief, then reweight it by how likely the evidence is under each hypothesis.

Quick Example

A disease test is 99% accurate; 1% of people have the disease. If you test positive, P(disease∣+)≈50%P(\text{disease}|+) \approx 50\% — not 99%, because the disease is rare.

Notation

P(A)P(A) is the prior, P(B∣A)P(B \mid A) is the likelihood, P(A∣B)P(A \mid B) is the posterior, and P(B)P(B) is the total evidence probability.

What This Formula Means

Bayes' theorem gives the posterior probability of a hypothesis given evidence: P(H∣E)=P(E∣H)⋅P(H)P(E)P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}.

Start with a prior belief, then reweight it by how likely the evidence is under each hypothesis.

Formal View

For events AA and BB with P(B)>0P(B) > 0: P(A∣B)=P(B∣A)⋅P(A)P(B)P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} where P(B)=P(B∣A)P(A)+P(B∣Ac)P(Ac)P(B) = P(B|A)P(A) + P(B|A^c)P(A^c) by the law of total probability.

Worked Examples

Example 1

medium
Email spam filter: P(spam)=0.3P(\text{spam})=0.3. The word 'free' appears in 80% of spam emails and 10% of legitimate emails. An email contains 'free'. Find P(spam∣free)P(\text{spam}|\text{free}) using Bayes' theorem.

Answer

P(spam∣free)≈0.774P(\text{spam}|\text{free}) \approx 0.774. There's a 77.4% chance the email is spam.

First step

1
Prior: P(spam)=0.3P(\text{spam})=0.3, P(legit)=0.7P(\text{legit})=0.7

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Example 2

hard
Drug testing: P(user)=0.05P(\text{user})=0.05. Test sensitivity P(+∣user)=0.99P(+|\text{user})=0.99. Specificity P(−∣non-user)=0.95P(-|\text{non-user})=0.95 (so P(+∣non-user)=0.05P(+|\text{non-user})=0.05). Find P(user∣+)P(\text{user}|+).

Example 3

medium
A box contains 40% red and 60% blue marbles. Red marbles are 'shiny' 30% of the time; blue marbles are shiny 10% of the time. A drawn marble is shiny. Find P(red∣shiny)P(\text{red}|\text{shiny}).

Common Mistakes

  • Treating P(H∣E)P(H|E) as equal to P(E∣H)P(E|H) - Bayes flips them, and the prior makes the two differ.
  • Ignoring the base rate (prior) - a rare condition keeps the posterior low even after strong evidence.
  • Using the wrong denominator - P(E)P(E) must total over ALL hypotheses (true and false), e.g. true positives plus false positives.

Why This Formula Matters

Real questions ask 'given a positive test, do I have the disease?' but data give you 'given the disease, how often does the test come back positive?' — Bayes is the only way to flip that, and ignoring the base rate (the prior) is the classic error behind wildly overstated test-result fears. It formalizes learning from evidence. Recognizing it by "Am I given P(E∣H)P(E|H) and a prior, and asked for the flipped P(H∣E)P(H|E)?" — rather than by familiar numbers — is what lets a student tell it apart from conditional probability and compound probability and law of total probability in a mixed problem set.

Frequently Asked Questions

What is the Bayes' Theorem formula?

Bayes' theorem gives the posterior probability of a hypothesis given evidence: P(H∣E)=P(E∣H)⋅P(H)P(E)P(H|E) = \frac{P(E|H) \cdot P(H)}{P(E)}.

How do you use the Bayes' Theorem formula?

Start with a prior belief, then reweight it by how likely the evidence is under each hypothesis.

What do the symbols mean in the Bayes' Theorem formula?

P(A)P(A) is the prior, P(B∣A)P(B \mid A) is the likelihood, P(A∣B)P(A \mid B) is the posterior, and P(B)P(B) is the total evidence probability.

Why is the Bayes' Theorem formula important in Math?

Real questions ask 'given a positive test, do I have the disease?' but data give you 'given the disease, how often does the test come back positive?' — Bayes is the only way to flip that, and ignoring the base rate (the prior) is the classic error behind wildly overstated test-result fears. It formalizes learning from evidence. Recognizing it by "Am I given P(E∣H)P(E|H) and a prior, and asked for the flipped P(H∣E)P(H|E)?" — rather than by familiar numbers — is what lets a student tell it apart from conditional probability and compound probability and law of total probability in a mixed problem set.

What do students get wrong about Bayes' Theorem?

The procedure for bayes' theorem is the easy part; the trap is treating P(H∣E)P(H|E) as equal to P(E∣H)P(E|H). Asking "Am I given P(E∣H)P(E|H) and a prior, and asked for the flipped P(H∣E)P(H|E)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Bayes' Theorem formula?

Before studying the Bayes' Theorem formula, you should understand: conditional probability, probability, sample space.

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