Venn diagrams and sets

Venn diagrams sort things into overlapping circles. They make it easy to count and to find probabilities with “and”, “or” and “not”.

Key facts

  • A ∩ B: in A and B (the overlap). A ∪ B: in A or B or both.
  • A′: not in A. ξ is everything.
  • Fill a Venn diagram from the middle out.
  • P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Worked examples

Example 1

Of 30 pupils, 18 play football, 12 play tennis and 5 play both. How many play neither?

  1. Both: 5.
  2. Football only: 18 − 5 = 13. Tennis only: 12 − 5 = 7.
  3. Neither: 30 − 13 − 5 − 7 = 5.

Answer: 5

Example 2

P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2. Find P(A ∪ B).

  1. 0.5 + 0.4 − 0.2 = 0.7

Answer: 0.7