Tennis Game

Look carefully at how the deuce might be similar to another situation.

First, we start with summing up the possible scenarios. Bing can win in three different scenarios:

1. Bing gets two points in a row, .
2. Bing gets a point, then Luuk gets a point and then Bing wins via a deuce.
3. Luuk gets a point, then Bing gets a point and then Bing wins via a deuce.

The whole (hard) trick in this question is to figure out how to handle the probability related to a deuce. It seems like the deuce can be a never ending summation of probabilities. However, the trick is to recognise that the deuce-situation is equal to when the game is 30-30. Therefore, the equations to solve this problem are as follows:

(1)  

(2)  

As stated above, the deuce-situation is equal to when the game is 30-30.

(3)  

(4)  

(5)  

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