Example

# 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.

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)