FRM Exam Part I · Fundamentals of Probability

Bayes' Theorem for FRM Part I: Prior, Likelihood, Posterior

Updated 11 October 2026 · Fact-checked

Bayes' theorem updates a prior probability after you see new evidence. Posterior = likelihood × prior ÷ total probability of the evidence. To solve a problem, define the events, list the priors and likelihoods, compute P(evidence) with the law of total probability, then divide. Check that posteriors sum to 1.

Understand Bayes' Theorem

Bayes' theorem answers a reversed question. You know how likely the evidence is if an event is true, for example P(warning signal | default). You want the opposite: P(default | warning signal). Bayes' rule converts one into the other.

Start with a prior: your probability for the event before seeing the evidence. Then observe the evidence. The likelihood tells you how probable that evidence is under each possible state. The posterior is your updated probability after weighing the prior against the likelihoods.

The key idea is that the evidence is judged against everything that could produce it. A signal that fires often for non-defaulters is weak evidence, even if it fires nearly always for defaulters. When the event is rare, most signals come from the large non-event group. This is why a very accurate test can still give a low posterior.

In risk work you see this with default prediction, fraud alerts, model flags, and rating or scorecard outputs. The denominator, P(evidence), comes from the law of total probability: add up prior × likelihood over every state. The states must be mutually exclusive and exhaustive, so that together they cover all outcomes.

Key formulas to remember

Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B); Valid when P(B) > 0. Rearranged: P(A ∩ B) = P(B | A) × P(A).
Bayes' theorem (two events): P(A | B) = P(B | A) × P(A) ÷ P(B); A is the hypothesis, B is the evidence. Needs P(B) > 0.
Law of total probability: P(B) = P(B | A) × P(A) + P(B | not A) × P(not A); Works because A and not A are mutually exclusive and exhaustive.
Bayes' theorem (several states): P(Aᵢ | B) = P(B | Aᵢ) × P(Aᵢ) ÷ Σⱼ [P(B | Aⱼ) × P(Aⱼ)]; The Aᵢ must be mutually exclusive and exhaustive. Posteriors sum to 1.
Odds form: Posterior odds = prior odds × likelihood ratio; Odds = p ÷ (1 − p). Likelihood ratio = P(B | A) ÷ P(B | not A). Convert odds back with p = odds ÷ (1 + odds).

How to solve Bayes' Theorem questions

Use this method for any Bayes question. It keeps the numbers organised and catches the usual traps.

1Name the events. Let A be the hypothesis (for example, the firm defaults) and B be the evidence (for example, the model flags it).
2Write the priors P(A) and P(not A). If the question gives only one, subtract from 1. If there are several states, list all of them and check the priors sum to 1.
3Write the likelihoods P(B | state) for every state. Make sure each is conditioned the right way round.
4Compute the joint probabilities P(B | state) × P(state) for each state.
5Add the joint probabilities to get P(B), the total probability of the evidence.
6Divide the joint probability of the state you want by P(B). This is the posterior.
7Sanity check: posteriors across all states sum to 1, and the posterior moved in the direction the evidence suggests.

Quickest way: Natural-frequency (imagine 100,000 cases) method

When to use it: Use it for two-state problems with rates given as percentages, when you want speed and few algebra errors.

Pick a large round population, such as 100,000.
Split it by the prior: the number in the event group and the number outside it.
Apply the likelihoods to each group to count how many show the evidence.
Posterior = evidence cases in the event group ÷ all evidence cases.
Match the answer to the closest option, then check the sign of the update.

Common mistakes in Bayes' Theorem

Treating P(B | A) as if it were P(A | B)
The two sound alike. A test with a 95% hit rate feels like a 95% chance the flagged firm defaults.
Fix: Write each probability with its condition explicitly. Ask which event is given and which is being asked about.
Ignoring the prior (base rate)
Candidates focus on the accuracy of the signal and forget how rare the event is.
Fix: Always start by writing P(A). A low prior pulls the posterior down, even with a strong likelihood.
Leaving out the false-positive term in the denominator
Candidates use only P(B | A) × P(A) as P(B).
Fix: P(B) must include every route to the evidence, including P(B | not A) × P(not A).
Using a specificity as a false-positive rate
Questions may give the rate of correct negatives, so P(B | not A) = 1 − that figure.
Fix: Read the wording carefully. False-positive rate = 1 − specificity.
Posteriors that do not sum to 1 in multi-state problems
The states are not exhaustive, or an arithmetic slip happened in the denominator.
Fix: Compute all posteriors, or at least confirm that the denominator equals the sum of all joint terms.

Worked examples

Example 1

A bank estimates that 2% of its corporate borrowers will default within a year. Its early-warning model flags 90% of borrowers who go on to default, and also flags 10% of borrowers who do not. A borrower is flagged. What is the probability that it defaults?

Show the solution

Let D = default and F = flagged. P(D) = 0.02, P(not D) = 0.98.
Likelihoods: P(F | D) = 0.90 and P(F | not D) = 0.10.
Joint with default: 0.90 × 0.02 = 0.018.
Joint with no default: 0.10 × 0.98 = 0.098.
P(F) = 0.018 + 0.098 = 0.116.
P(D | F) = 0.018 ÷ 0.116 = 0.1552, about 15.5%.

Answer: About 15.5%. The flag raises the default probability from 2% to roughly 15.5%, but most flagged borrowers still do not default.

Example 2

A risk manager believes a fund manager is one of three types: skilled with probability 0.20, average with probability 0.50, or unskilled with probability 0.30. The probability of beating the benchmark in a year is 0.70 for skilled, 0.50 for average and 0.30 for unskilled. The manager beats the benchmark this year. What is the posterior probability that the manager is skilled?

Show the solution

Let S, A, U be the types and W the event of beating the benchmark. Priors: 0.20, 0.50, 0.30 (sum = 1).
Joint for skilled: 0.70 × 0.20 = 0.14.
Joint for average: 0.50 × 0.50 = 0.25.
Joint for unskilled: 0.30 × 0.30 = 0.09.
P(W) = 0.14 + 0.25 + 0.09 = 0.48.
P(S | W) = 0.14 ÷ 0.48 = 0.2917, about 29.2%.

Answer: About 29.2%. One good year lifts the probability of skill from 20% to roughly 29%.

Exam tips

Look for wording like 'given that', 'flagged', 'tests positive' or 'after observing'. These signal a posterior is wanted.
Before calculating, estimate the answer. If the prior is tiny and the false-positive rate is not tiny, the posterior should be low.
Write the joint probabilities in a small list. Exam options often include the likelihood itself or the numerator alone as traps.
Keep four decimal places until the final division, then round to the precision of the options.
If a question gives joint probabilities in a table, skip the formula: the posterior is the cell divided by the column or row total.

Practice questions from Fundamentals of Probability

Bayes' Theorem in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Bayes' Theorem: frequently asked questions

What are prior, likelihood and posterior in Bayes' theorem?

The prior is your probability for the event before the evidence. The likelihood is the probability of the evidence if the event is true. The posterior is the updated probability after seeing the evidence, found by weighing the likelihood against the prior.

How is Bayes' theorem used in finance and risk?

It updates beliefs about default, fraud, manager skill or model state as new data arrives. For example, it converts a model's hit rate into the probability that a flagged borrower actually defaults. It also underlies Bayesian approaches to estimating parameters.

Do I need to memorise Bayes' theorem for FRM Part I?

Yes, but the formula is short. Remember it as joint probability divided by total probability of the evidence. Practise with the denominator expanded using the law of total probability.

Why is my posterior so low when the test is accurate?

When the event is rare, the non-event group is huge. Even a small false-positive rate on a large group produces many false alarms, which dilute the true signals.