Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 A person is chosen at random from the group shown. Find P(A or B).
Show worked solution
The group: 12 + 3 + 9 + 6 = 30. Inside the circles: 12 + 3 + 9 = 24, so P = \(\dfrac{24}{30}\) = \(\dfrac{4}{5}\). \(\dfrac{9}{10}\) = \(\dfrac{27}{30}\) adds the circles (15 + 12) and counts the overlap twice; \(\dfrac{7}{10}\) = \(\dfrac{21}{30}\) leaves out the overlap; \(\dfrac{1}{10}\) is "A and B".
2 In a class of 32, 20 students like maths, 15 like art and 8 like both. Fill in the Venn diagram: how many like maths only, art only, and neither?
Show worked solution
Maths only = 20 − 8 = 12. Art only = 15 − 8 = 7. Inside the circles: 12 + 8 + 7 = 27, so neither = 32 − 27 = 5.
3 In a survey of 120 people, 70 drink tea, 55 drink coffee and 20 drink neither. How many drink both?
Show worked solution
At least one: 120 − 20 = 100. So 70 + 55 − both = 100, and both = 125 − 100 = 25.
Level 2 · Core
4 Some cells of this two-way table are missing. Which Venn diagram regions match it?
Show worked solution
A, not B = 15 − 6 = 9. B, not A = 14 − 6 = 8. Neither = 40 − 9 − 6 − 8 = 17. So A only 9, both 6, B only 8, outside 17. "A only 15 … B only 14" puts the totals in the only regions; "B only 14" uses the column total for B only; "outside 23" = 40 − 9 − 8 forgets the overlap.
5 A person is chosen at random from the group shown. Find P(A or B, but not both), in simplest form.
Show worked solution
The group: 18 + 12 + 20 + 10 = 60. A only or B only: 18 + 20 = 38, so P = \(\dfrac{38}{60}\) = 19/30.
6 For two events, P(A) = 0.6, P(B) = 0.5 and P(A and B) = 0.25. Find P(neither A nor B).
Show worked solution
P(A or B) = 0.6 + 0.5 − 0.25 = 0.85, so P(neither) = 1 − 0.85 = 0.15.
Level 3 · Challenge
7 Of 50 students, 30 study Art (A), 25 study Music (M), 5 study neither, and x study both.1 x = 10.
2 P(Art only) = \(\dfrac{2}{5}\).
3 P(not Art) = \(\dfrac{1}{2}\).
Which statements are correct?
2 P(Art only) = \(\dfrac{2}{5}\).
3 P(not Art) = \(\dfrac{1}{2}\).
Which statements are correct?
Show worked solution
(30 − x) + x + (25 − x) + 5 = 50 gives 60 − x = 50, so x = 10 (1 ✓). Art only = 30 − 10 = 20, and \(\dfrac{20}{50}\) = \(\dfrac{2}{5}\) (2 ✓). Not Art = Music only + neither = 15 + 5 = 20, so P = \(\dfrac{20}{50}\) = \(\dfrac{2}{5}\), not \(\dfrac{1}{2}\) (3 ✗). 1 and 2.
8 In a group of 50, n(A) = 24 and n(B) = 20. The number in neither set is twice the number in both. Find, in simplest form, P(both) and P(exactly one of A, B).
Show worked solution
(24 − x) + x + (20 − x) + 2x = 50 gives 44 + x = 50, so x = 6. P(both) = \(\dfrac{6}{50}\) = 3/25. A only = 18 and B only = 14, so exactly one = 32 and P = \(\dfrac{32}{50}\) = 16/25. Check: 18 + 6 + 14 + 12 = 50 ✓.
9 There are 80 students in a year: 35 are boys, and 30 students wear glasses. A student is chosen at random, and P(boy and wears glasses) = \(\dfrac{1}{5}\). How many girls do not wear glasses?
Show worked solution
Boys with glasses = \(\dfrac{1}{5}\) × 80 = 16. Girls with glasses = 30 − 16 = 14. Girls = 80 − 35 = 45, so girls without glasses = 45 − 14 = 31.