A number is chosen at random from 1 to 20. A: the number is even; B: the number is a multiple of 3. Find P(A ∩ B), P(A ∪ B), P(Ā) and P(A ∩ B̄).

1. 1 · Write what is given and what is to be found S = {1, 2, ..., 20}, n(S) = 20. To find: four probabilities.
2. 2 · Draw the Venn diagram Draw a rectangle S with two overlapping circles A and B. Put 6, 12 and 18 in the overlap first.
3. 3 · Write the formulas P(E) = n(E)/n(S); P(Ā) = 1 − P(A); P(A ∩ B̄) = P(A) − P(A ∩ B).
4. 4 · List the events
   • A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}, n(A) = 10
   • B = {3, 6, 9, 12, 15, 18}, n(B) = 6
   • A ∩ B = {6, 12, 18}, n(A ∩ B) = 3
5. 5 · Work out the probabilities
   • P(A ∩ B) = 3/20
   • A ∪ B has 10 + 6 − 3 = 13 numbers, so P(A ∪ B) = 13/20
   • P(Ā) = 1 − 10/20 = 1/2
   • P(A ∩ B̄) = 10/20 − 3/20 = 7/20
6. 6 · Check and box the answer P(A ∩ B) = 3/20, P(A ∪ B) = 13/20, P(Ā) = 1/2, P(A ∩ B̄) = 7/20. Check: 2, 4, 8, 10, 14, 16 and 20 are the 7 even numbers that are not multiples of 3.

Scoring tip: Fill the overlap of the Venn diagram first, then the rest of each circle. Counting from the diagram avoids counting a number twice.