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Year 11 Mathematics · Conditional probability

Name: ______________________

Date: ____________

1. What does P(A | B) mean?
- ☐ (a) The probability of A or B
- ☐ (b) The probability of A given that B has happened
- ☐ (c) The probability of A divided by B
- ☐ (d) The probability that neither A nor B happens
2. In a group of 50 students, 30 play sport, and 18 of those also have a pet. A student who plays sport is chosen at random. What is the probability they have a pet? Give a fraction in simplest form.
3. A bag holds 4 red and 6 green marbles. Two are taken without replacement. What is the probability that both are green? Give a fraction in simplest form.
4. Which pair of events is independent?
- ☐ (a) Tossing a head on a coin and rolling a 6 on a die
- ☐ (b) Drawing a king, then drawing another king from the same pack without replacement
- ☐ (c) Taking a red counter from a bag, then another red counter without putting the first back
- ☐ (d) It raining today and the ground being wet today
5. In a group of 30 people, 12 like cats, 15 like dogs and 5 like both. A person who likes dogs is chosen at random. What is the probability they also like cats? Give a fraction in simplest form.
6. For two events, P(A) = 0.3, P(B) = 0.6 and P(A and B) = 0.25. Are A and B independent?
- ☐ (a) Yes, because 0.3 + 0.6 is not 1
- ☐ (b) No, because P(A) × P(B) = 0.18, which is not equal to 0.25
- ☐ (c) Yes, because 0.25 is less than 0.3
- ☐ (d) It is impossible to tell
7. If P(A and B) = 0.12 and P(B) = 0.3, find P(A | B).
8. A bag has 5 red and 3 blue counters. One is taken out and not replaced. Given the first counter was red, what is the probability that the second is blue?
- ☐ (a) 38
- ☐ (b) 37
- ☐ (c) 27
- ☐ (d) 1556

Answer sheet

1. The probability of A given that B has happened — The vertical line is read 'given'.
2. 3/5 — Restrict to the 30 sport players: P(pet | sport) = 18/30 = 3/5.
3. 1/3 — P(green, green) = 6/10 × 5/9 = 30/90 = 1/3.
4. Tossing a head on a coin and rolling a 6 on a die — The coin has no effect on the die, so the probability of rolling a 6 stays 16 whatever the coin shows.
5. 1/3 — Restrict to the 15 dog lovers, of whom 5 also like cats: 5/15 = 1/3.
6. No, because P(A) × P(B) = 0.18, which is not equal to 0.25 — Independent events satisfy P(A and B) = P(A) × P(B). Here 0.3 × 0.6 = 0.18 ≠ 0.25, so the events are dependent.
7. 0.4 — P(A | B) = P(A and B) ÷ P(B) = 0.12 ÷ 0.3 = 0.4.
8. 37 — After a red is removed, 7 counters remain and 3 of them are blue, so the probability is 37.