Probability of Combined Events
How to Use the addition rule for probability
Use the addition rule to find the probability that one event OR another happens.
Before you start
- Finding the probability of a single event
- Adding and subtracting fractions with a common denominator
- Knowing what 'mutually exclusive' means
- Finding the overlap (intersection) of two events
When to use it
Use the addition rule to find the probability that one event OR another happens.
The steps
- Decide whether the two events can happen together or are mutually exclusive.
- If mutually exclusive, add their probabilities: P(A) + P(B).
- If they can overlap, subtract the overlap: P(A) + P(B) − P(A and B).
- Compute each probability first, then combine.
- Check the total does not exceed 1.
Worked example
A jar holds 20 sweets: 8 red, 5 green and 7 yellow. One sweet is taken at random.
Find the probability that it is red or green.
- Decide the type: a sweet is exactly one colour, so 'red' and 'green' cannot happen together, they are mutually exclusive.
- Because they are mutually exclusive, use P(red) + P(green) with nothing to subtract.
- Compute each probability first: P(red) = 8/20, P(green) = 5/20.
- Combine: 8/20 + 5/20 = 13/20.
- Check the total does not exceed 1: 13/20 = 0.65 ≤ 1, so it is valid.
A second example, with a twist
This time the two events can happen together (one number is in both lists), so you must subtract the overlap. A number is chosen at random from 1 to 20.
Find the probability that it is a multiple of 4 or a multiple of 5.
- Decide the type: multiples of 4 are 4, 8, 12, 16, 20; multiples of 5 are 5, 10, 15, 20. The number 20 is in both, so the events can overlap.
- Because they overlap, use P(A) + P(B) − P(A and B).
- Compute each first: P(multiple of 4) = 5/20, P(multiple of 5) = 4/20, P(both, i.e. a multiple of 20) = 1/20.
- Combine: 5/20 + 4/20 − 1/20 = 8/20 = 2/5.
- Check the total does not exceed 1: 2/5 = 0.4 ≤ 1, so it is valid.
Formulae you may need
Formula pages
Practise this in a KBAT problem
Frequently asked questions
How do I tell if two events are mutually exclusive?
Ask whether any single outcome can satisfy both events at once. If none can, they are mutually exclusive and you simply add.
If at least one outcome is in both, they overlap and you must subtract P(A and B) once.
Why do I subtract the overlap?
Any outcome that satisfies both events is counted once in P(A) and again in P(B), so it is counted twice. Subtracting P(A and B) once removes that double count, leaving each outcome counted exactly one time.
What happens if I forget to subtract the overlap?
Your total will be too large because the shared outcomes are double-counted, and it may even come out greater than 1. A probability above 1 is impossible, so it is a clear warning sign that the events overlapped and you skipped the subtraction.