A gambler starts with $\$k$ and repeatedly bets $\$1$ on a fair coin flip -- heads they win $\$1$, tails they lose $\$1$. The game ends when their bankroll hits $\$n$ (they win) or $\$0$ (they're ruined). What is the probability the gambler reaches $\$n$ before going broke?

Gambler's Ruin on a Fair Coin

Probability · Medium · Free problem

A gambler starts with $\$k$ and repeatedly bets $\

$ on a fair coin flip -- heads they win $\

$, tails they lose $\

$. The game ends when their bankroll hits $\$n$ (they win) or $\$0$ (they're ruined). What is the probability the gambler reaches $\$n$ before going broke?

Open the full interactive solver, hints, and worked solution →