FRM Exam Part I · Fundamentals of Probability

Independent Events and Conditional Probability for FRM Part 1

Updated 11 October 2026 · Fact-checked

Conditional probability is the chance of event A given that event B has occurred: P(A | B) = P(A ∩ B) ÷ P(B), for P(B) > 0. Events are independent if P(A | B) = P(A), which means P(A ∩ B) = P(A) × P(B). Find the joint probability first, then divide.

Understand Independent Events and Conditional Probability

A marginal (unconditional) probability is the chance of an event with no extra information, such as P(a bond defaults this year). A joint probability is the chance that two events both happen, written P(A ∩ B). A conditional probability is the chance of A once you know B has happened. Knowing B shrinks your sample space to B only.

The formula follows from that idea. Of all outcomes where B happens, you want the share where A also happens. So P(A | B) = P(A ∩ B) ÷ P(B). Rearranged, it gives the multiplication rule: P(A ∩ B) = P(A | B) × P(B). This works in either order, so P(A ∩ B) = P(B | A) × P(A) as well.

Two events are independent if knowing one tells you nothing about the other. Then P(A | B) = P(A), and the joint probability is just the product: P(A ∩ B) = P(A) × P(B). In finance, defaults of two firms in unrelated industries are sometimes treated as independent. In a crisis they usually are not, which is why correlation matters.

Do not confuse independent with mutually exclusive. Mutually exclusive events cannot happen together, so P(A ∩ B) = 0. If both have positive probability, they are strongly dependent: if A happens, B certainly did not. Independent events with positive probabilities can always happen together.

To get a marginal probability from conditional ones, use the law of total probability: weight each conditional probability by the chance of its condition. This is the bridge to Bayes' theorem.

Key formulas to remember

Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B); Valid only when P(B) > 0.
Multiplication rule: P(A ∩ B) = P(A | B) × P(B) = P(B | A) × P(A); Works for any events, dependent or not.
Independence: P(A ∩ B) = P(A) × P(B), equivalently P(A | B) = P(A); For events with non-zero probability, any one of these conditions implies the others.
Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); If mutually exclusive, the last term is 0.
Law of total probability: P(A) = Σ P(A | Bᵢ) × P(Bᵢ); The Bᵢ must be mutually exclusive and exhaustive.
Mutually exclusive events: P(A ∩ B) = 0; Two mutually exclusive events with positive probabilities are not independent.

How to solve Independent Events and Conditional Probability questions

Use this routine for any question on joint, marginal or conditional probability.

1Define the events with letters and write down every probability given, marking each as marginal, joint or conditional.
2Identify what is asked. The word 'given', 'if' or 'conditional on' means the condition goes after the bar.
3If the question says the events are independent, use P(A ∩ B) = P(A) × P(B). Do not assume independence otherwise.
4If you need a joint probability from a conditional one, multiply: P(A | B) × P(B).
5If you need a conditional probability, compute the joint probability and divide by the probability of the condition.
6For a marginal probability across scenarios, use the law of total probability.
7Check the answer lies between 0 and 1 and that the condition sits in the denominator.

Quickest way: Imagine 1,000 cases

When to use it: When the question gives percentages and asks for a conditional probability.

Pick a convenient population, such as 10,000 firms.
Use the marginal and conditional probabilities to count how many fall in each cell.
Restrict to the condition group only.
Divide the count with both events by the count of the condition group.
Test independence by checking whether P(A | B) equals P(A).

Common mistakes in Independent Events and Conditional Probability

Dividing by the wrong probability, giving P(B | A) instead of P(A | B).
The wording 'A given B' is read backwards under time pressure.
Fix: The condition is always the denominator. Write the event after 'given' first.
Treating mutually exclusive events as independent.
Both ideas sound like 'unrelated'.
Fix: Mutually exclusive means P(A ∩ B) = 0. Independent means the joint equals the product. With positive probabilities they cannot both hold.
Multiplying marginals when events are dependent.
The product rule is simple and memorised first.
Fix: Use P(A ∩ B) = P(A | B) × P(B) unless independence is stated or proven.
Adding probabilities of events that can overlap without subtracting the joint.
The mutually exclusive version is remembered as the general rule.
Fix: Use P(A) + P(B) − P(A ∩ B) unless the events are mutually exclusive.
Assuming P(A | B) = P(B | A).
The two look symmetric.
Fix: They share the numerator P(A ∩ B) but have different denominators. Convert using Bayes' theorem.

Worked examples

Example 1

Two unrelated firms, X and Y, have one-year default probabilities of 2% and 5%. Assuming defaults are independent, what is the probability that at least one defaults?

Show the solution

P(X) = 0.02, P(Y) = 0.05.
Independence gives P(X ∩ Y) = 0.02 × 0.05 = 0.001.
P(X ∪ Y) = 0.02 + 0.05 − 0.001 = 0.069.

Answer: 6.9%

Example 2

In a portfolio, 8% of loans are in default. Of loans in default, 75% are in a sector that had a downturn. Overall, 20% of loans are in sector downturn. What is the probability a loan is in default given it is in a sector downturn?

Show the solution

Let D = default, S = sector downturn. P(D) = 0.08, P(S | D) = 0.75, P(S) = 0.20.
Joint: P(D ∩ S) = P(S | D) × P(D) = 0.75 × 0.08 = 0.06.
Conditional: P(D | S) = 0.06 ÷ 0.20 = 0.30.
Check independence: P(D | S) = 0.30 ≠ P(D) = 0.08, so the events are dependent.

Answer: 30%

Exam tips

Read the condition carefully. Many wrong options are the reversed conditional probability.
Check whether the question states independence. If not, you need a conditional probability or a joint probability given.
If an option equals P(A) × P(B) and the events are not independent, it is a likely trap.
Sanity-check: a conditional probability can exceed the marginal one when the events are positively related.
Practise the 10,000-case table; it avoids formula errors.

Practice questions from Fundamentals of Probability

Independent Events and Conditional Probability in other exams

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

CA FoundationConditional Probability

Independent Events and Conditional Probability: frequently asked questions

What is the conditional probability formula in FRM Part I?

P(A | B) = P(A ∩ B) ÷ P(B), where P(B) > 0. You find the joint probability and divide by the probability of the condition.

What is the difference between independent and mutually exclusive events?

Independent events satisfy P(A ∩ B) = P(A) × P(B), so one gives no information about the other. Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. With positive probabilities, mutually exclusive events are dependent.

How do I calculate the joint probability of independent events?

Multiply their marginal probabilities. For events with probabilities 0.3 and 0.4, the joint probability is 0.12.

What is the difference between unconditional and conditional probability?

An unconditional probability uses no extra information. A conditional probability uses the knowledge that another event has occurred, so the relevant sample space is smaller.