Conditional probability tree diagram

3 Jimmy and Alan are playing a tennis match against each other. The winner of the match is the first player to win three sets. Jimmy won the first set and Alan won the second set. For each of the remaining sets, the probability that Jimmy wins a set is

• 0.7 if he won the previous set,
• 0.4 if Alan won the previous set.

It is not possible to draw a set.

1. Draw a probability tree diagram to illustrate the possible outcomes for each of the remaining sets.
2. Find the probability that Alan wins the match.
3. Find the probability that the match ends after exactly four sets have been played.

7 Jenny has six darts. She throws darts, one at a time, aiming each at the bull's-eye. The probability that she hits the bull's-eye with her first dart is 0.1 . For any subsequent throw, the probability of hitting the bull's-eye is 0.2 if the previous dart hit the bull's-eye and 0.05 otherwise.

1. Illustrate the possible outcomes for her first, second and third darts on a probability tree diagram.
2. Find the probability that
   (A) she hits the bull's-eye with at least one of her first three darts,
   (B) she hits the bull's-eye with exactly one of her first three darts.
3. Given that she hits the bull's-eye with at least one of her first three darts, find the probability that she hits the bull's-eye with exactly one of them. Jenny decides that, if she hits the bull's-eye with any of her first three darts, she will stop after throwing three darts. Otherwise she will throw all six darts.
4. Find the probability that she hits the bull's-eye three times in total.