How Bankroll Size Affects Your Chances

25 October 1997

By Alan Krigman

Among the weakest moves solid citizens make in casinos is playing with too little money. Or, equivalently, wagering too high for their bankrolls. Bettors who thus tempt fate typically believe that gambling is mainly or entirely luck. That it's mostly or all a matter of being in the right place at the right time.

Undercapitalized gambling can certainly pay. Everyone's heard the same anecdotes - factual, flatulent, and fictional - about folks who found fortune on their final or only dimes. And, of course, the chance of winning any particular bet or set of bets doesn't depend on how much money is in reserve. But the amount of a gambling stake relative to the size of the bets being made affects the odds of a successful session.

This aspect of betting theory is highlighted in the accompanying table. Entries give probability of winning and expected duration of a session for various combinations of bankroll and win goal. Data are for flat craps pass line and single-zero roulette even-money bets; these have single-trial probabilities of winning of 49.3 and 48.6 percent, respectively. Bankrolls and goals are in bet units, so numbers like 5 and 10 represent $50 and $100 for $10 players.

The table shows how bankroll impacts session odds for the brigades of bettors who won't quit with less than some fixed profit, regardless of the amount with which they start. They view goals in absolute dollars rather than percent return on money at risk.

Here's an example. Say that Sammy, Tammy, and Ignatz play craps. They make only $5 pass line bets with no odds. Each of the trio is hoping for $50 profit. However, bankrolls differ: $25 for Sammy, $50 for Tammy, and $100 for Ignatz.

For the same $50 goal, the table indicates how likelihood of attainment grows with bankroll. Here, Sammy has 29 percent chance to hit the mark. Tammy has 43 percent chance to succeed. Ignatz is looking at 57 percent chance of a score.

Separately, the table suggests that underfunding reduces expected number of plays before bettors reach their win goals or lose everything. Consider the craps crew, still betting $5 on pass and looking to win $50. With a $25 stake Sammy can expect 49 decisions, boom or bust. Tammy's $50 anticipates 99 rounds, good or bad. And Ignatz should go 207 come-outs on his $100, win or lose.

Expected number of plays has a psychological impact on chances of success. If Sammy makes $50 in only 49 rounds, he may think he's on a roll and keep playing; if he loses his $25 after this short a time, he may dig into his fanny pack for more. Conversely, whether Ignatz finally tastes sweet victory or bitter defeat after grinding out 207 rounds, he'll likely take a hike.

The table reveals another interesting effect. If players set their sights in terms of return on a stake rather than fixed profit, their chances for equivalent multipliers fall as bankroll rises relative to bet size. Back again to the dauntless dice doyens dropping $5 on the line but now determined to double their dough. Sammy has 46 percent chance to earn $25 on his $25 stake. Tammy has a 43 percent shot at doubling $50. And Ignatz has only 36 percent probability of netting $100 on his $100 buy-in.

Relationships governing bankroll, bet size, and odds of successful sessions are complex - and influenced by the house's inherent edge and the chance of winning each wager. No facile rule covers every situation. Especially because personal response to gains and losses cloud the mathematical clarity. In general, though, limiting assets increases players' vulnerability. Sumner A Ingmark, a bard oft bereft of bankroll heft, said it this way:

Gambling's for folks who well can afford it,
Others should take their money and hoard it.