A player serving in tennis has two chances to get a serve into play. If the first serve is out, the player serves again. If the second serve is also out, the player loses the point. Observe the probabilities based on four years of the Wimbledon Championship. P(1st serve in)=0.59 P(win point | 1st serve in)=0.73 P(2nd serve in | 1st serve out)=0.86 P(win point | 1st serve out and 2nd serve in)=0.59 Make a tree diagram for the results of the two serves and the outcome, win or lose, of the point. The branches in your tree have different numbers of stages depending on the outcome of the first serve. What is the probability that the serving player wins the point? Report your answer to four decimal places.

Wimbledon Championship. P(1st serve in)=0.59 P(win point | 1st serve in)=0.73 P(2nd serve in | 1st serve out)=0.86 P(win point | 1st serve out and 2nd serve in)=0.59

If P(1st serve in)=0.59 then, P(1st serve out) = 1-0.59=0.41

If P(win point | 1st serve in)=0.73 then P(loss point | 1st serve in) = 1-0.73=0.27

If P(2nd serve in | 1st serve out)=0.86 then P(2nd serve out | 1st serve out) = 1-0.86=0.14

If P(win point | 1st serve out and 2nd serve in)=0.59 then P(loss point | 1st serve out and 2nd serve in)=1-0.59=0.41

Therefore, the required diagram is shown below:

Now calculate the probability that the serving player wins the point.

[tex]P(Win)=[0.59\times 0.73]+[0.41\times 0.86\times 0.59]\\P(Win)=0.638734\approx 0.6387[/tex]

Hence, the winning probability is 0.6387