Gambler's ruin

A small edge, a finite bankroll. Find out how likely you are to hit zero before you hit your goal.

Chance of winning each bet47.4%

Starting money$100

Walk away at$200

Size of each bet$10

Exact chance of reaching the goal25.9%

Exact chance of going broke74.1%

Expected number of bets92

Of these players38 won, 112 brokeout of 150

Top edge: the goal of $200. Bottom edge: broke. Lines that touch either edge stop there. The chart spans 367 bets.

Each bet is only slightly against you (47.4% to win), yet starting with $100 and betting $10 at a time, you reach $200 just 25.9% of the time. The casino doesn't need to win every bet: a small edge, repeated about 92 times, grinds the player down.

Should you bet big or small?

Chance of reaching $200 from $100 for every bet size. When the odds are against you, fewer and bigger bets give the edge less time to work. When the odds favour you, the opposite is true.

Try this

Pick roulette and drop the bet to $1. How does a 47.4% chance per spin become almost certain ruin?
Switch to the card counter's edge and keep bets small. Why do casinos ban players with a 1% edge?
On a fair coin, raise the goal to $1,000. What happens to your chance as the casino gets richer?

The math

P(\text{reach } N) = \frac{1 - (q/p)^{i}}{1 - (q/p)^{N}}

$p$: chance of winning one bet
$q$: chance of losing one bet, 1 − p
$i$: starting money, counted in bets
$N$: goal, counted in bets (for p = q, the chance is i / N)

Ask the Loop about this Next: The Kelly criterion