Natasha and Allison are playing a tennis match where the winner must win 2 sets

Question

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 0.6. However, the probability she wins the second set is only 0.5. And if a third set is needed, the probability that Natasha wins the third set is only 0.35. 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 40% 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? The radio broadcast of AM station KTBC can be heard in the neighboring towns of Waterville and Newport. Of the total number of radio listeners in Waterville and Newport, 80% live in Waterville and 20% live in Newport. Radio station KTBC has 15% of the market in Waterville and 40% of the market in Newport. (a) If a random listener in Waterville or Newport is surveyed, what is the probability that the listener will be tuned in to station KTBC? (b) If you know that the listener is indeed listening to station KTBC, what then is the conditional probability that the listener lives in Newport?

Explanation / Answer

Solution:(1)

(a) P(N wins) =
P(N N) = .6 * .5 = 0.3 +
P(N A N) = .6 * .5 * .35 = 0.105 +
P(A N N) = .4 * .5 * .35 = 0.07
and sum of those = 0.475

(b) "given that Allison wins the first set"
means everything else is irrelevant
and P(Natasha wins) = P(N wins 2nd set & N wins 3rd set) = 0.5 * 0.35 = 0.175

(c) Similarly "given that Natasha wins first set" we go straight to the remaining sets
and P(Allison wins) = P(A wins 2nd & A wins 3rd) = .5 * .65 = 0.325

(d) P (3 sets needed) =
1st two sets are A - N or N - A winning and we have
.4 * .5 + .6 * .5 = 0.5

You can check that by computing P(N N) and P(A A) which would be 1 - 0.5 and in fact

.6 * .5 + .4 * .5 = 0.5 which is right

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