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 0.7. 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 0.3.

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?

Explanation / Answer

probability that Natasha wins the match =P(wins first and second +wins loses second and wins third+loses first and wins last two) =0.7*0.5+0.7*0.5*0.3+0.3*0.5*0.3=0.5

b)

probability Allison wins first match =P(wins wins +wins loses wins +wins loses loses)=0.3*0.5+0.3*0.5*0.7+0.3*0.5*0.3=0.3

hence probability that Natasha, instead, ends up winning the match given Allison wins first set

=0.3*0.5*0.3/0.3 =0.15

c)

similarly conditional probability that Allison, instead, ends up winning the match given Natasha wins first set =0.5*0.7 =0.35

d)probability that 3 sets will be played =1-P(either Natasha or Allison wins first two set)

=1-(0.7*0.5+0.3*0.5)=1-0.5 =0.5

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