# Thread: Expected number of turns in single player coin flipping game

1. ## Expected number of turns in single player coin flipping game

Similar to my other topic, again I play a coin flipping game.

This time I start with zero points. I repeatedly flip a coin. Heads = I get one point, tails = I lose one point. My score can also be negative.

Question: what's the expected number of turns I have to take, to reach 3 points?

2. ## Re: Expected number of turns in single player coin flipping game

Defining $T_{a\rightarrow b}$ as the number of turns until reaching b points (starting with a points), I think $\mathbb{E}(T_{0\rightarrow 3}) = 3\mathbb{E}(T_{0\rightarrow 1})$ and $\mathbb{E}(T_{0\rightarrow 1})=\mathbb{E}(T_{n\rightarrow n+1})\ \forall n$.
So basically it boils down to calculating $\mathbb{E}(T_{n\rightarrow n+1})$, the expected number of turns to reach one point more than I currently have.
I can write the recurrence relation $\mathbb{E}(T_{n\rightarrow n+1}) = \frac{1}{2} + \frac{1}{2}\mathbb{E}(T_{n-1\rightarrow n+1})$ but since there's no boundaries to the score, this seems to bring me nowhere?