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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 anguish-ing

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

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

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.

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 Sumner
A Ingmark, saw such situations:


Casino perks are an assurance,

For bettors with enough endurance.

Decisions needed for 90% 95%, and 99%
confidence in success



95% 99%

craps, “no ten”




craps, natural on the come-out




blackjack, natural




craps, 12




0-roulette, single spot




00-roulette, single spot




big six, 45-to-1 winner




1-joker video poker, 5-of-a-kind




7-card stud poker, royal




Carib Stud or let-it-ride, royal 1,496,080 1,946,446 2,992,161
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