23.4 Combined events and tree diagrams — Practise

1. A fair coin is tossed twice. Find the probability of two heads.

Answer: $\frac12\times\frac12=\frac14$.

2. A fair coin is tossed twice. Find the probability of exactly one head.

Answer: HT and TH give $\frac24=\frac12$.

3. A fair die is rolled twice. Find the probability of at least one six.

Answer: $1-\frac{25}{36}=\frac{11}{36}$.

4. A bag has 3 red and 2 blue counters. A counter is replaced after each pick. Find the probability of red then blue.

Answer: $\frac35\times\frac25=\frac6{25}$.

5. A fair coin is tossed three times. Find the probability of three tails.

Answer: $\frac18$.

6. A fair coin is tossed twice. Find the probability of at most one head.

Answer: TT, HT and TH: $\frac34$.

7. A spinner has probability $\frac14$ of red. It is spun and a coin is tossed. Find the probability of red and heads.

Answer: $\frac18$.

8. A fair die is rolled twice. Find the probability of no sixes.

Answer: $\frac{25}{36}$.

9. A fair coin is tossed twice. Find the probability of exactly one tail.

Answer: HT and TH, so $\frac12$.

10. An event has probability $\frac13$ on each independent trial. Find the probability it happens on all three trials.

Answer: $\frac1{27}$.

11. A bag has 4 red and 3 blue counters. One counter is drawn and not replaced. Find the probability of red then blue.

Answer: After red, 3 of the 6 remaining counters are blue: $\frac47\times\frac36=\frac27$.