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Longshots Can Be Attractive, But Not as Steady Bets

27 August 1996

Longshots are bets with exhilaratingly high payoffs and excruciatingly low probabilities. Although gaming gurus often belittle them, these wagers have undeniable appeal.

The primary argument against longshots is that the house usually has too great an edge. Returns, while monetarily large, are disproportionately small compared with the odds of winning. For instance, any specific three-of-a-kind at sic bo pays 180-to-1; however, the odds against winning are 216-to-1. The difference between 216 and 180 gives the casino a big advantage.

Occasionally, with progressive jackpots, the payoff grows high enough to shift theoretical advantage to the player. Here, the naysayers contend that the likelihood of winning is negligible. An example would be a $225,000 jackpot for a $1 side bet in Caribbean stud. At this prize level, players have a mathematical edge; however, the odds are a staggering 72,192-to-1 against receiving a $22,500 non-royal straight flush and an even worse 649,739-to-1 against getting a $225,000 royal.

These arguments are balanced by the subjective quality of risk. All math aside, people think risk is low when the payoff is high and the amount bet seems trivial. If a video poker machine pays $10 million for a $1.25 bet on an ascending royal straight flush in spades, usurious house edge and odds against winning of 311,875,199-to-1 don't necessarily dampen the appeal.

There's a trap, though. It's set by not realizing how remote a chance is associated with a longshot. And, it's sprung by treating longshots, not as occasional small bets, but as series of wagers that can add up to substantial loss. The trap is most insidious when solid citizens overreach their financial or psychological limits, then continue to play - desperately trying to recover by hitting a jackpot - wanting to believe that the "law of averages" makes them due for a win.

Sure, stories are legion about players who squeezed their last pennies from those ever-so-convenient bank machines, dropped them into the slots, and became instant millionaires. But, gambling's not about what rumor says happened to someone you don't know. It's about what the laws of probability say could happen to you.

Here's one way to envision the futile effect of frequent play on the likelihood of hitting a longshot. How many trials must a player bankroll to achieve a reasonable chance of winning? The accompanying table gives some representative answers and shows how numbers escalate with odds. As an example of interpreting the table, a player drawing a five-card poker hand bet has to anticipate 438,147 trials for the probability of a royal flush to exceed 50 percent and 1,455,490 trials to surpass 90 percent.

Here's another way to picture the ineffectual influence of frequent play on the chance of hitting a longshot. How many trials are needed before the statistics characterizing cumulative performance match the theoretical probabilities of the individual bets? I've used the "law of large numbers" - admittedly taking liberties with the logic - to get the estimates in the last column of the table. My figures assume a 90 percent chance that a series is within 10% of the theoretical single-trial probability. For a 99 percent chance that the series will be within 1 percent of the theoretical probability, multiply the indicated numbers by 1000. The requirements turn out to be huge: for instance, 17,000 trials before a craps player can be 90 percent confident that a series of bets on the 11 will be within half a percent of the statistically-correct distribution.

So, it's OK to indulge your fantasy of striking it rich with a small outlay by taking a longshot now and then. Just don't make a career of it; the investment grows a lot faster than the chance of winning. As the immortal poet Sumner A Ingmark wrote:

The occasional longshot,
That could put me in clover,
'Though 'till now's been a wrong shot,
When it's over, it's over.

Numbers of trials required before longshots can be
expected to conform with theoretical probability profiles
trials for a
50% chance
of winning
trials for a
90% chance
of winning
trials to
conform with
law of
large numbers
3 or 11 at craps
number at roulette
non-royal straight flush, nothing wild
royal straight flush, nothing wild

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.