Natasha and Allison are playing a tennis match where the winner must win 2 sets in order to win the match.

Natasha starts strong but tires quickly. The probability that Natasha wins the first set is 0.65. However, the probability she wins the second set is only 0.55. And if a third set is needed, the probability that Natasha wins the third set is only 0.3.

Put all this information into a tree diagram to answer the following questions.

(a) What is the probability that Natasha wins the match?

(b) If Allison wins the first set, what is the conditional probability that Natasha, instead, ends up winning the match?

(c) If Natasha wins the first set, what is the conditional probability that Allison, instead, ends up winning the match?

(d) What is the probability that 3 sets will be played?

Carla is trying to pass a competency exam. Each time she takes the exam she has a 20% chance of passing, and she is allowed a maximum of two attempts.

Draw a tree diagram to represent her attempts to pass the exam. Answer the following questions.

(a) How many outcomes does your tree show?

(b) What is the probability she will eventually pass the exam?

(c) What is the probability she will take the exam twice?