23.6 Conditional probability — Practise

Work through these independently. Every question gives you a chance to check what you understand.

H Of 20 students who cycle, 15 wear helmets. Find the probability a cyclist wears a helmet.
H Of 18 bus users, 7 bring lunch. Find $P(\text{lunch}\mid\text{bus})$ .
H In a group of 30, 12 are in A and 8 are in both A and B. Find $P(B\mid A)$ .
H A bag has 4 red and 3 blue counters. Given that a red counter was removed, find the probability the next counter is blue.
H There are 16 students in a team. Six are left-handed, and 4 of the left-handed students wear glasses. Find the probability a student wears glasses, given that the student is left-handed.
H In a group, 14 girls attend a club and 9 of them walk home. Find the probability a club attendee walks home, given the attendee is a girl.
H A fair die is rolled. Given the result is even, find the probability it is greater than 3.
H In a group, 24 are in A and 10 are in both A and B. Find $P(A\cap B\mid A)$ .
H A bag has 5 red and 3 blue counters. One red is removed without replacement. Find the probability the second counter is red, given the first was red.
H A fair die is rolled. Given the result is greater than 2, find the probability it is odd.
H In a group of 30 students, 18 study French and 10 study both French and Spanish. Find $P(\text{Spanish}\mid\text{French})$ . Which representation makes the restricted group clearest: a two-way table or a Venn diagram? Give a reason, then find the probability.

Answers and concise solutions

1. Of 20 students who cycle, 15 wear helmets. Find the probability a cyclist wears a helmet.

Answer: $\frac34$ .

2. Of 18 bus users, 7 bring lunch. Find $P(\text{lunch}\mid\text{bus})$ .

Answer: $\frac7{18}$ .

3. In a group of 30, 12 are in A and 8 are in both A and B. Find $P(B\mid A)$ .

Answer: $\frac23$ .

4. A bag has 4 red and 3 blue counters. Given that a red counter was removed, find the probability the next counter is blue.

Answer: After removal, 3 of 6 are blue: $\frac12$ .

5. There are 16 students in a team. Six are left-handed, and 4 of the left-handed students wear glasses. Find the probability a student wears glasses, given that the student is left-handed.

Answer: Among the 6 left-handed students, 4 wear glasses: $\frac46=\frac23$ .

6. In a group, 14 girls attend a club and 9 of them walk home. Find the probability a club attendee walks home, given the attendee is a girl.

Answer: $\frac9{14}$ .

7. A fair die is rolled. Given the result is even, find the probability it is greater than 3.

Answer: Among 2, 4 and 6, two exceed 3: $\frac23$ .

8. In a group, 24 are in A and 10 are in both A and B. Find $P(A\cap B\mid A)$ .

Answer: $\frac5{12}$ .

9. A bag has 5 red and 3 blue counters. One red is removed without replacement. Find the probability the second counter is red, given the first was red.

Answer: Four red remain among seven counters: $\frac47$ .

10. A fair die is rolled. Given the result is greater than 2, find the probability it is odd.

Answer: The restricted results are 3, 4, 5, 6; two are odd, so $\frac12$ .

11. Method choice: table or Venn diagram

Answer: Either works, because both show the overlap of 10 inside the 18 French students. The restricted total is 18, so $P(\text{Spanish}\mid\text{French})=\frac{10}{18}=\frac59$ .