Probability of not losing the game Jake flips a biased coin with a probability of $p=0.7$ for $H$, until the coin shows $H$. The number of coin flips then determine the amount of coins Jake is allowed to pull out of a jar, with replacement, that contains 100 coins out of which 25 are real and 75 a fake. For each real coin Jake pulls out of the jar, he receives $10.
Assuming that Jake pays $20 for participating in this game, what is the probability that he didn't lose money?

Let $X$ denote the number of flips. 
There will be no loss if at least $2$ real coins are pulled out and
denoting this event with $E$ we find:
$$P\left(E\mid X=k\right)=1-\left(\frac{3}{4}\right)^{k}-k\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^{k-1}$$
So that:
$$P\left(E\right)=\sum_{k=1}^{\infty}P\left(E\mid X=k\right)P\left(X=k\right)=$$$$\sum_{k=1}^{\infty}\left[1-\left(\frac{3}{4}\right)^{k}-k\left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^{k-1}\right]\left(\frac{3}{10}\right)^{k-1}\left(\frac{7}{10}\right)$$
