How Long Must You Play Before You Can Expect to Win?

Some folks patronize casinos to dabble. A couple of bucks at a machine, a few bets worth of buy-in at a table, and quit when they lose the money or score a hit. They know the laws of chance are against them, but are hoping to get lucky.

Other people are more serious about their action. They not only show up at their favorite joints frequently, but approach the games with big enough bankrolls to weather the normal cold spells and give themselves fair shots at winning sessions. These solid citizens anticipate ups and downs. Good and bad games. Exhilarating and anguishing trips. Many believe that triumph entails time as much as luck, that resolve reaps rewards. They tend to wonder less whether fortune will come smiling, than when.

You don't have to be a Nikolai Ivanovich Lobatchevskii to know that, provided an event is possible, the odds it will happen at least once increase the more often you try. Picture this for yourself by imagining an unbiased coin. Flip once and the coin can land heads or tails. Betting on heads, your chance would be 50 percent. Instead, flip twice and ask the chance of at least one heads. Two flips can yield HH, HT, TH, or TT - each equally likely. Heads occurs at least once in three of the four potential outcomes, so the probability is 3/4 or 75 percent. Flip three times and you can get HHH, HHT, HTH, THH, HTT, THT, TTH, or TTT - again, each equally likely. Now, seven of the eight prospects include at least one heads so the probability is 7/8 or 87.5 percent. Four flips give 15 out of 16 or 93.75 percent; five flips yield 31 out of 32 or 96.875 percent, and so on. The probability approaches 100 percent, although it never quite gets there. By flipping sufficiently often, you can become arbitrarily confident but never 100 percent certain of success.

The same idea applies to real casino wagers. Only, for many situations, it's more practical to invert the question. That is, for some particular bet, how many tries do you need to achieve a specified degree of confidence you'd win at least once?

Say you play double-zero roulette, betting on red. You're using a Martingale system, starting with $5 and doubling-up whenever you lose. You reason that red will eventually hit, recouping your losses and paying a $5 profit. It takes four spins for 90 percent confidence of at least one win, five spins for 95 percent, and seven spins for 99 percent. Assuming you have the wherewithal for seven rounds, $635, you'd succeed 99 percent of the time you try this. But you'll still face a 1 percent chance of disaster.

Maybe you fancy video poker, indulge heavily, and have faith that over an extended period you'll get your share of royals. The chance of a royal in a jacks-or-better game, following optimum strategy, is around one out of 40,000. You'd have to play almost 92,500 hands to be 90 percent confident of hitting at least one royal. At 15 hands per minute - a relatively fast pace - this is just over 100 hours. You'd need more than 120,000 hands, 130 or so hours, for 95 percent confidence. And 185,000 hands, roughly 200 hours, would be necessary for 99 percent confidence.

The accompanying table gives the numbers of decisions needed for 90, 95, and 99 percent confidence in achieving at least one success in a variety of other casino bets.

Decisions needed for 90% 95%, and 99% confidence in success
bet	90%	95%	99%
craps, "no ten"	2	3	4
craps, natural on the come-out	9	12	18
blackjack, natural	47	62	95
craps, 12	82	106	163
0-roulette, single spot	84	109	168
00-roulette, single spot	86	112	173
big six, 45-to-1 winner	123	160	246
1-joker video poker, 5-of-a-kind	23,917	31,117	47,834
7-card stud poker, royal	71,242	92,688	142,484
Carib Stud or let-it-ride, royal	1,496,080	1,946,446	2,992,161

There's an exception to the 100 percent certainty rule. True, no amount of play can guarantee a win. But something like 163 bets on boxcars at craps, 173 rounds straight-up at roulette, or 246 wagers on the whopper at a big six wheel will surely earn you a commiseration comp to the all-you-can-eat buffet. Here's how the sanguine songster, Sumner A Ingmark, saw such situations:

Casino perks are an assurance,
For bettors with enough endurance.