What is the probability of winning given amount of games?

Question

Probability of winning in a single game equals p*100%. How many at least games have to be played to, with given probability β, win at least m times?

I think negative binomial distribution should be used here, but i do not know how to work it out.

Answer 1

So you have a game which you win with probability $p$. You want to find the min $K$ such that: $$\mathbb{P}[\text{win at least $m$ times out of $K$}] \geq \beta$$However, this can be written as: $$\sum_{i=m}^K {K \choose i} p^i (1-p)^{K-i}\geq \beta$$which you can ask a computer to solve given $p, m$ and $\beta$.