13E - The law of total probability

The law of total probability

The law of total probability allows us to find the the total probability of an event which may occur via several distinct conditional events. The law states:

From the formula we can see that the event A can occur given B has occurred or given that B has not occurred (B') - in either case the event A has occurred. Thus, the probability of A occurring is given by the addition of the two aforementioned events.

Roger Federer

Throughout Roger Federer's professional tennis career he has served 62% of first serves in. If he gets his first serve in he wins 77% of the time. On his second serve he wins 57% of the points. What is the total probability that Federer will win any given point?

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13E - Exercises:

Question 1:

Three current Men's singles players have 10 or more Grand Slam titles and together have held the world number 1 position since 2004: Roger Federer, Rafael Nadal and Novak Djokovic. All players are in contention to go down as the greatest of all time. Bruce McAvaney wants to know the total probability of each of these players winning a point during their own service game. Using the law of total probability and the information in the table below, rank the three players from highest to lowest probability of winning a point on their serve (assume that double faults have a negligible impact):

Question 2:

Serena Williams is a Women rewriting the history of Women's Singles in Tennis. In the 2015 Wimbledon Final the probability that Serena won any point on her serve was 0.5822. When Serena's first serve went in, she won 78% of the point. On her second serve, Serena was winning only 35% of points. What was the percentage of first serve that Serena served in for the match?