Tennis Game

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

1. Bing gets two points in a row, $0.4*0.4$.
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: \begin{equation}
P(Bing \; | \; 30-30) = 0.4*0.4 + 0.4*0.6*P(Bing \; | \; Deuce) + 0.6*0.4*P(Bing \; | \; Deuce)
\end{equation} \begin{equation}
P(Bing \; | \; 30-30) = 0.4*0.4 + 2*0.4*0.6*P(Bing \; | \; Deuce)
\end{equation} As stated above, the deuce-situation is equal to when the game is 30-30. \begin{equation}
P(Bing \; | \; 30-30) = 0.4*0.4 + 2*0.4*0.6*P(Bing \; | \; 30-30)
\end{equation} \begin{equation}
0.52*P(Bing \; | \; 30-30) = 0.16
\end{equation} \begin{equation}
P(Bing \; | \; 30-30) = 30.7\%
\end{equation}