Events are mutually exclusive if they cannot happen at the same time, such as rolling a 2 and rolling a 5 on one die. For mutually exclusive events, P(A or B) = P(A) + P(B). If a set of mutually exclusive events covers every possible outcome, their probabilities add up to 1, which lets you find a missing probability.
The OR rule
- Mutually exclusive: cannot happen together
- P(A or B) = P(A) + P(B), for mutually exclusive events
- Exhaustive mutually exclusive events add up to 1
- If events can happen together, do not simply add
What does mutually exclusive mean?
Two events are mutually exclusive if they cannot both happen at once. On a single roll of a die, “rolling a 3” and “rolling an even number” are mutually exclusive, because 3 is odd. “Rolling an even number” and “rolling more than 3” are not, because 4 and 6 are both.
The OR rule
For mutually exclusive events, the probability of one or the other is found by adding: P(A or B) = P(A) + P(B). If P(red) = 0.27 and P(blue) = 0.38, then P(red or blue) = 0.65.
Exhaustive events
If a set of mutually exclusive events covers every possible outcome, the events are exhaustive and their probabilities add up to 1. This is how to find a missing value in a probability table: subtract the known probabilities from 1.
When not to add
If events can happen together, adding counts the overlap twice. For a die, P(even) = 1/2 and P(more than 3) = 1/2, but P(even or more than 3) is not 1: the outcomes 2, 4, 5 and 6 give 4/6 = 2/3. Use a list, a Venn diagram or a sample space instead.
Probability tables
Many questions give a table of probabilities with one value missing, or with an expression such as x and 2x. Use the fact that the total is 1 to form an equation, then solve it.
Worked examples
Example 1. A fair die is rolled. Find P(a prime number or a 6).
- Primes on a die: 2, 3 and 5. These cannot also be 6, so the events are mutually exclusive.
- P(prime) + P(6) = 3/6 + 1/6.
Answer: 4/6 = 2/3.
Example 2. P(red) = 0.27 and P(blue) = 0.38 for a spinner. Find P(red or blue).
- The colours are mutually exclusive, so add.
Answer: 0.65.
Example 3. A bag contains only red, blue and white beads. P(red) = 0.15 and P(blue) = 0.48. Find P(white).
- The events are exhaustive: 1 − 0.15 − 0.48.
Answer: 0.37.
Example 4. A, B and C are the only outcomes of a game. P(A) = 0.16, P(B) = x and P(C) = 3x. Find x.
- 0.16 + x + 3x = 1, so 4x = 0.84.
Answer: x = 0.21.
Example 5. Explain why P(even or more than 3) on one die roll is not 1/2 + 1/2.
- 4 and 6 are both even and more than 3, so the events can happen together.
- The correct outcomes are 2, 4, 5 and 6.
Answer: The events are not mutually exclusive; the probability is 4/6 = 2/3.
Common mistakes
- Adding probabilities of events that can happen together. Only add for mutually exclusive events.
- Multiplying for OR. OR means add (for mutually exclusive events); AND means multiply (for independent events).
- Forgetting that exhaustive events total 1. Use 1 minus the known probabilities.
- Missing an algebraic expression in the total. Include every term, such as x and 3x, when forming the equation.
- Writing a total greater than 1. If the probabilities add to more than 1, something has gone wrong.
How mutually exclusive events are examined
Questions on mutually exclusive events usually come with a probability table to complete, on Foundation papers for both GCSE and iGCSE. Questions ask you to:
- find P(A or B) for mutually exclusive events
- find a missing probability in a table
- solve for an unknown such as x in a probability table
- explain whether two events are mutually exclusive
The AND and OR rules worksheet includes answers, and the probability revision engine generates more.
This builds on basic probability. For events that happen together, see the AND and OR rules.
Frequently asked questions
What are mutually exclusive events?
Events that cannot happen at the same time.
How do you find P(A or B) for mutually exclusive events?
Add the probabilities: P(A) + P(B).
Do mutually exclusive events always add to 1?
Only if together they cover every possible outcome.