A level-readyAll subjects · Mathematics · WST01 (S1)
Venn diagrams and the addition rule
Venn diagrams show how events overlap. They make the addition rule, mutually exclusive events and 'only' regions easy to see.
Notation
- A ∩ B: A and B (the overlap).
- A ∪ B: A or B or both (everything inside either circle).
- A': not A (everything outside A).
The rectangle is the whole sample space, with total probability 1.
The addition rule
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
You subtract the overlap because it was counted twice. If A and B are mutually exclusive (cannot happen together), P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
Filling in a Venn diagram
Start with the innermost region (the overlap, or the triple overlap with three events) and work outwards, subtracting as you go. Remember the region outside all circles. With frequencies, divide by the total to get probabilities.
Example: P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15 gives A only = 0.35, B only = 0.25, neither = 1 - 0.75 = 0.25.
Three events
Read regions carefully: 'football and tennis' includes the students who do all three; 'football and tennis only' does not.
| Region | Number of students |
|---|---|
| Football only | 10 |
| Tennis only | 6 |
| Swimming only | 8 |
| Football and tennis only | 4 |
| Football and swimming only | 5 |
| Tennis and swimming only | 3 |
| All three | 2 |
| None of the three sports | 12 |
A Venn diagram of 50 students' sports, written as a table of regions
Worked example
P(A) = 0.5, P(B) = 0.4 and P(A ∩ B) = 0.15. Find P(A ∪ B) and P(neither).
- P(A ∪ B) = 0.5 + 0.4 - 0.15 = 0.75
- P(neither) = 1 - 0.75
Answer: 0.75 and 0.25
Check yourself
- P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15. Find P(A ∪ B).
Show answer
0.75. 0.5 + 0.4 - 0.15 = 0.75.
- P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.15. Find P(A ∩ B'), the probability of A but not B.
Show answer
0.35. A only = 0.5 - 0.15 = 0.35.
- P(A) = 0.6, P(B) = 0.3 and P(A ∪ B) = 0.75. Find P(A ∩ B).
Show answer
0.15. Rearrange: P(A ∩ B) = 0.6 + 0.3 - 0.75 = 0.15.
- A and B are mutually exclusive with P(A) = 0.25 and P(B) = 0.4. Find P(A ∪ B).
Show answer
0.65. No overlap: 0.25 + 0.4 = 0.65.
- Using the table of regions, find the probability that a randomly chosen student plays football. Give your answer as a fraction in its simplest form.
Show answer
21/50. Football regions: 10 + 4 + 5 + 2 = 21, so 21/50.
- Using the table of regions, find the probability that a randomly chosen student plays exactly one of the three sports. Give your answer as a fraction in its simplest form.
Show answer
12/25. Only regions: 10 + 6 + 8 = 24, so 24/50 = 12/25.
- What does P(A ∩ B) = 0 tell you?
- The events are independent
- The events are mutually exclusive
- A is certain to happen
- The events are complements of each other
Show answer
The events are mutually exclusive. They cannot happen together.
Sources
Last updated 7 November 2026. Spotted a mistake? Tell us.