FRM Exam Part I · Fundamentals of Probability

Events, Mutually Exclusive and Exhaustive Events Explained

Updated 11 October 2026 · Fact-checked

An event is a set of outcomes from a sample space. Mutually exclusive events cannot happen together, so P(A ∩ B) = 0. Exhaustive events cover the whole sample space, so their probabilities sum to 1 if they are also exclusive. For any two events, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Understand Events, Mutually Exclusive and Exhaustive Events

A sample space is the set of all possible outcomes of a random process. Roll one die and the sample space is {1, 2, 3, 4, 5, 6}. An event is any subset of the sample space, such as 'the roll is even' = {2, 4, 6}.

Two events are mutually exclusive (disjoint) if they have no outcome in common. They cannot both occur on the same trial. 'Roll a 1' and 'roll a 6' are mutually exclusive. Their intersection is empty, so P(A ∩ B) = 0.

Events are exhaustive if together they cover the whole sample space. At least one of them must occur. Their union is the sample space. 'Roll at most 3' and 'roll at least 3' are exhaustive, but not mutually exclusive, because 3 is in both.

A set of events that is both mutually exclusive and exhaustive is a partition. Exactly one event occurs, and the probabilities add to 1. A bond ending the year as 'defaulted' or 'not defaulted' is a simple example. The complement of A, written Aᶜ, is everything not in A, and A and Aᶜ always form a partition.

Do not confuse mutually exclusive with independent. Exclusive events are strongly dependent: if one occurs, the other cannot. Independent events satisfy P(A ∩ B) = P(A) × P(B). If both events have non-zero probability, they cannot be both exclusive and independent.

Key formulas to remember

Complement rule: P(Aᶜ) = 1 − P(A); Use it when 'at least one' is easier as 1 minus 'none'.
Addition rule (general): P(A ∪ B) = P(A) + P(B) − P(A ∩ B); Works for any two events. Subtract the overlap so it is not counted twice.
Addition rule (mutually exclusive): P(A ∪ B) = P(A) + P(B); Valid only when P(A ∩ B) = 0.
Mutually exclusive events: P(A ∩ B) = 0; They cannot occur together.
Exhaustive events: P(A₁ ∪ A₂ ∪ … ∪ Aₙ) = 1; Together they cover the sample space.
Partition: P(A₁) + P(A₂) + … + P(Aₙ) = 1; Holds when the events are both mutually exclusive and exhaustive.
Independence: P(A ∩ B) = P(A) × P(B); A different idea from exclusivity.
Union of three events: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A∩B) − P(A∩C) − P(B∩C) + P(A∩B∩C); Add singles, subtract pairs, add the triple.

How to solve Events, Mutually Exclusive and Exhaustive Events questions

Use this method for any question on events, unions, intersections and complements.

1Write down the sample space or the process in one line.
2Define each event with a letter and note what it contains.
3Decide whether the events are mutually exclusive, exhaustive, independent, or none. Check the wording and the numbers given.
4Choose the rule: addition rule for 'or', complement for 'at least one' or 'not', and the product rule only for independent events.
5Check whether the overlap P(A ∩ B) is given. If the events are exclusive it is 0. If not, you must have it or derive it.
6Substitute the numbers and compute.
7Check that the answer lies between 0 and 1 and that it is at least as large as each single probability when you computed a union.

Quickest way: Count by region, not by formula

When to use it: When a question gives P(A), P(B) and an overlap or union, and asks for another region.

Picture two overlapping circles inside a box of total 1.
Fill the overlap first, then A only = P(A) − overlap and B only = P(B) − overlap.
Neither = 1 − (A only + overlap + B only).
Read the region the question asks for.

Common mistakes in Events, Mutually Exclusive and Exhaustive Events

Treating mutually exclusive as independent
Both ideas sound like 'unrelated'.
Fix: Exclusive means P(A ∩ B) = 0. Independent means P(A ∩ B) = P(A) P(B). With non-zero probabilities, exclusive events are dependent.
Adding P(A) + P(B) for events that overlap
The simple sum is learned first.
Fix: Use the general addition rule and subtract P(A ∩ B) unless the events are exclusive.
Assuming exhaustive events sum to 1
Students forget the events may overlap.
Fix: The sum is 1 only if the events are also mutually exclusive. Otherwise the sum is at least 1.
Forgetting that a union means 'A or B or both'
Everyday 'or' often means one or the other.
Fix: In probability, ∪ includes the overlap. 'Exactly one' is P(A) + P(B) − 2P(A ∩ B).
Computing 'at least one' by long addition
Students do not use the complement.
Fix: Compute 1 − P(none). It is faster and has fewer errors.

Worked examples

Example 1

A risk manager reviews two counterparties. P(A defaults in a year) = 0.04, P(B defaults) = 0.06, and P(both default) = 0.01. What is the probability that at least one defaults, and the probability that neither defaults?

Show the solution

P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
P(A ∪ B) = 0.04 + 0.06 − 0.01 = 0.09.
P(neither) = 1 − P(A ∪ B) = 1 − 0.09 = 0.91.

Answer: At least one defaults with probability 0.09. Neither defaults with probability 0.91.

Example 2

A bond's rating at year end is one of three mutually exclusive and exhaustive outcomes: upgrade, unchanged, downgrade. P(upgrade) = 0.10 and P(downgrade) = 0.15. What is P(unchanged), and P(rating changes)? Are 'upgrade' and 'unchanged' independent?

Show the solution

The three outcomes form a partition, so the probabilities sum to 1.
P(unchanged) = 1 − 0.10 − 0.15 = 0.75.
Rating changes = upgrade or downgrade. These are exclusive, so P = 0.10 + 0.15 = 0.25.
Check independence: P(upgrade ∩ unchanged) = 0, but P(upgrade) × P(unchanged) = 0.10 × 0.75 = 0.075, which is not 0.
So they are not independent.

Answer: P(unchanged) = 0.75 and P(rating changes) = 0.25. Upgrade and unchanged are mutually exclusive, so they are not independent.

Exam tips

Read the wording: 'or' means union, 'and' means intersection, 'not' means complement.
If the question says exclusive, drop the overlap term. If it does not say so, do not assume it.
Use the complement for any 'at least one' question.
Sanity check: a union can never be smaller than either event, and no probability can exceed 1.
Expect conceptual options that confuse exclusive with independent. Eliminate them with the P(A ∩ B) test.

Practice questions from Fundamentals of Probability

Events, Mutually Exclusive and Exhaustive Events: frequently asked questions

What is the difference between mutually exclusive and independent events?

Mutually exclusive events cannot occur together, so P(A ∩ B) = 0. Independent events satisfy P(A ∩ B) = P(A) P(B). If both have non-zero probability, they cannot be both.

What are exhaustive events with an example?

Exhaustive events together cover every possible outcome. On a die roll, 'even' and 'odd' are exhaustive. 'At most 4' and 'at least 4' are also exhaustive, but they overlap at 4.

How do I find the probability of the union of two events?

Use P(A ∪ B) = P(A) + P(B) − P(A ∩ B). If the events are mutually exclusive, the last term is 0 and you just add.

Can events be exhaustive but not mutually exclusive?

Yes. 'Roll at most 3' and 'roll at least 3' cover all outcomes but share the outcome 3. Their probabilities sum to more than 1.