Tennis game

General
Bing and Luuk are playing tennis. The game is in a tie, 30-30. The probability of Luuk winning a point (so for example, from 15 to 30, or 30 to 40) is equal to 6/10. What is the probability that Bing wins the game?

Hint

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

Answer

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 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)