The Higher Your Win Goal, The Less Chance You'll Reach It

Rules for leaving a gaming table or machine, for instance upon reaching a win goal or loss limit, are like parallel parking instructions. The books spell it out but nobody follows the step-by-step directions. After a few casino visits or curbside maneuvers, folks either get a "feel" for how to do it right, or just keep draining their resources or denting their Roadmasters.

So, I'll skip the money management laundry list and go right to some numbers that tie together win goals, loss limits, and the chances a player will hit the one before the other. Numbers - I hasten to note - that surprised but didn't delight me when I finished the calculations, and may have a similar impact on you.

I'll simplify things by examining three "even-money" bets. a) A hypothetical "fair" coin-matching game where participants have 50 percent chance to win and 50 percent chance to lose. b) A craps pass line bet with no odds where solid citizens have 49.3 percent chance to win and 50.7 percent chance to lose. c) A bet on red at single-zero roulette where players have 48.6 percent chance to win and 51.4 percent chance to lose. The house advantage is zero in the fair game, 1.41 percent on the craps bet, and 2.70 percent at roulette. All three cases exhibit low volatility, with fluctu-ation per round close to one unit and skew essentially zero.

Table I shows the probabilities of doubling a bankroll before going broke, making these bets one unit at a time. Here's how to interpret the data. Say players start with 20 units - for instance $100 with $5 bets. The chance they'll double their money before losing it is 50 percent in a fair game, 36 percent on craps line bets with no odds, and 25 percent on red at single-zero roulette. Players determined to more than double their money have less than the indicated chance for success; those content to quit before doubling their stakes have a greater chance.

Table II shows the amount players have a 50 percent chance of winning before hitting various loss limits for each wager. Here's how to interpret the data. Craps players making flat line bets have 50 percent chance to win 17 units before busting with 30-unit stakes, 22 units with 50-unit stakes, and 27 units with 100-unit stakes. For the same bankrolls, decreasing or increasing win goal raises or lowers the chance for success, respectively.

What's so surprising - and disappointing - about these results? Three things: 1) Small increases in house advantage sharply lower the likelihood of achieving any win goals. 2) High probabilities of success imply very low win goals. 3) Big increases in bankroll boost the amounts and likelihoods of wins only slightly.

More volatile games or betting strategies - those with higher-ratio/less-frequent payoffs or progressive wagers - would yield different outcomes. Such situations are subject to large fluctuations and have skews that tend to balance a few substantial wins against multiple small losses. During a typical playing session, the normal bankroll swings experienced in volatile games can swamp the effect of house advantage and quickly put players over the top or in deep holes. General conclusions about higher win goals being more difficult to achieve remain valid, but large bankrolls become more important owing to the need to outride normal downswings.

Either way, low volatility or high, the immortal Sumner A Ingmark wistfully worded the wisdom of the wager when he waxed:

When gambling goals are realistic,
Success rests not on forces mystic.

TABLE I Probability of Players Doubling Their Money Before Losing Their Bankrolls
bankroll (bet units)	fair game	craps flat line bet	single-zero roulette red
10	50%	43%	37%
20	50	36	25
30	50	30	26
40	50	24	10
50	50	20	06

TABLE II Win Goals With 50 Percent Probability of Achievement for Various Bankrolls (all entries are in bet units)
bankroll	fair game	craps flat line bet	single-zero roulette red
10	10	8	6
20	20	13	9
30	30	17	11
40	40	20	12
50	50	22	12
100	100	27	13