Suppose a gambler with infinite bankroll has a target of winning 10 dollars. He wins/loses $\$1$ with probabilities $0.48=p$ and $0.52=q respectively. What is the probability that he meets the target?
The answer using the usual methods is (p/q)^n = (12/13)^{10}.$
By a rather devious process, I have arrived at a combinatorial formula, $ \sum_{k=n}^{\infty}\frac{n}{k}{2k-n-1\choose k-n}p^kq^{(k-n)}, $ I get the same results, but can it be proved that the results will be identical?