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# Can you win at video poker without hitting any biggies?

20 December 2010

In some casino games, most notably those involving machines, a single bet can pay various amounts depending on the particular winning outcome of a coup. Jacks-or-better video poker is a good example. Pretend a sophisticated player follows the optimum hold-discard strategy for a particular version of this game with a high return percentage. For each possible winning hand, the accompanying table lists the payoff per unit bet, the chance of being hit, and the contribution to the overall return. Enquiring minds who prefer to think about the harsh reality of house advantage rather than the sweet reverie of player return will want to know that the two values are complementary. The house advantage or edge is 100 percent minus the return.

Hand              Payout    Probability    Return
Royal flush          800        0.0025%     1.98%
Straight flush        50        0.0109%     0.55%
Four of a kind        25        0.2363%     5.91%
Full house             9        1.1512%    10.36%
Flush                  6        1.1015%     6.61%
Straight               4        1.1229%     4.49%
Three of a kind        3        7.4449%    22.33%
Two pair               2       12.9279%    25.86%
High pair              1       21.4585%    21.46%
Nothing                0       54.5434%     0.00%
Total                         100.0000%    99.54%

The Total Return shown for the indicated game and strategy means that, averaged over lots of action, players get back 99.54 percent of the dough they feed into the machine. Not of the stake with which they started but of the gross amount placed at risk, including recycled earnings. The house keeps the remaining 0.46 percent – that's \$0.46 per \$100 wagered. In the short term, certainly over the span of one session or casino visit, solid citizens can finish ahead – sometimes way ahead. They can also lose, even their entire stakes, more often than anyone cares to imagine. The reason for this disparity is that in random processes like casino games, the actual frequencies at which results occur only equal those predicted by literally applying the laws of probability by sheer coincidence. The figures can be way off when the number of trials is small. As instances increase, results obtained get proportionately – if not numerically – closer to the theoretical values. You can get an idea of how many bets are needed for the theoretical projections to be at all reliable by examining the chances of the desirable hands at the high end of the list.

The probability of a royal flush is 0.0025 percent. Stated as a fraction, this is 1/40,000 – or, in words of a single syllable, one chance out of 40,000. So, using the arithmetic artifice of a statistically correct "cycle," a person would have to undergo 40,000 rounds before having much of a prayer that the game would yield one Royal. At the reasonably fast clip of a spin every five seconds, that's 55.5 hours. A person who visits casinos occasionally, for instance once a month, would need about a year of four-to-five-hour sessions to get this much time at the button.

A Royal could easily fail to appear in 40,000 rounds. In such a series of disappointing stints, were all other winning hands to occur at frequencies close to the nominal probability rates, overall return would drop by the 1.98 percent contribution of the Royal. It would therefore be 97.56 percent. Edge would accordingly rise to 2.44 percent. At \$1 a shot, expected loss would be 2.44 percent of \$40,000, or \$976. (Care to chance this waiting for an \$800 jackpot?) Similarly, the probability of a non-Royal straight flush is 0.0109 percent – one out of 9,174. After 10,000 rounds, a person might be unhappy but shouldn't be surprised not to have gotten either type of straight flush. Effective return is then 97.01 percent, a \$299 loss on a \$10,000 gross wager. Thankfully, four of a kind, which contributes a hefty 5.91 percent to return, develops more often. With a probability of 0.2363 percent, it's expected on the average of once every 423 rounds. That would be one or two per hour at one spin every five seconds. Playing for an hour and getting no straight flushes or quads but having all the other results close to the theoretical proportions, a person would have received about 91.11 percent return, and be down by an amount equivalent to an 8.89 percent edge . No straight flushes but one quad would yield the 97.02 percent return and cut the loss to the equivalent of 2.98 percent of the gross wager. No straight flushes but two quads would provide a 102.92 percent return, a profit of 2.92 percent of the handle.

If you jump on a machine with the intent of risking a few bucks in the hopes of catching a modest but quick profit, you can succeed without any of these biggies. At video poker, two pairs here or there, maybe a combination of trips, straights, and especially flushes lets you cash out and buy a little doo-dad on the bosses' nickel at the overpriced gift shop near the bus lobby. Of course, quads, a regular straight flush, or a Royal could always pop – one out of 40,000 in a random game doesn't necessitate 40,000 rounds before it appears – really making your day.

But you can see from the figures why a long session in which none of the three biggest hands appear is not unusual, and can grind you down, despite the experts telling you that the overall return on the game is close to 100 percent. It's precisely as the ever-popular punters' poet, Sumner A Ingmark, poignantly – perhaps profoundly – penned:

Grand Canyon was sculpted out not by explosion,
But over the eons, by inchmeal erosion.