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 s 0.6. However, the probability she wins the second set is only 0.5. And if a third set needed, the probability that Natasha wins the third set only 0.4. 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 Alison, instead, ends up winning the match? (d) what is the probability that 3 sets will be played?

Explanation / Answer

(a)
In order to win the match Natasha has to win 2 sets
WW(W+L); WLW; LWW
Hence required probability = 0.6*0.5*1 + 0.6*0.5*0.4 + 0.4*0.5*0.4 = 0.5

(b)
If Allison wins the first set, possible combinations are (In terms of Natasha Win or Lose)
LWW, LL(W+L), LWL
Probability that Allison wins the first set is 0.4*0.5*0.4 + 0.4*0.5*1 + 0.4*0.5*0.6 = 0.4
In this there is only one way where Natasha wins i.e. LWW whose probability is 0.4*0.5*0.4 = 0.08

Hence required conditional proability is = 0.08/0.4 = 0.2

(c)
Natasha wins the first set: WW(W+L); WLW; WLL, probability = 0.6*0.5*1 + 0.6*0.5*0.4 + 0.6*0.5*0.6 = 0.6
Allison wins : WLL, probability = 0.6*0.5*0.6 = 0.18

Hence required conditional probability = 0.18/0.6 = 0.3

(d)
Three sets will be played when WLW, LWL, WLL, LWW
Probability = 0.6*0.5*0.4 + 0.4*0.5*0.6 + 0.6*0.5*0.6 + 0.4*0.5*0.4 = 0.5

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