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Would the bosses be giving away the casino with 10s-or-better video poker?

22 August 2011

Pretend your friendly neighborhood casino just installed a platoon of multi-game slot machines, and you notice that one of them includes a 10s-or-better video poker option. You know enough about video poker to realize that the probabilities associated with various final hands aren’t arbitrary, as they are for the winning levels on reel-type slots. Assuming you either follow the optimum strategy for the payback schedule in question or use some common sense about familiar poker rankings, the chances are determined principally by the composition of a standard card deck. So a game on which 10s-or-better yields a payback has a greater probability of nominally successful rounds – a higher hit rate – than its jacks-or-better counterpart.

Can you take this to mean that the bosses have somehow goofed and are offering a gamble with a return percentage higher than normal, possibly over 100 percent and accordingly giving solid citizens an advantage? Or, as is often true in the wonderful albeit wacky world of wagering, is more happening behind that smoke and those mirrors than first appears? And, if so, what?

It’s possible, of course, that a mistake has been made in setting up the game and none of the bigwigs has noticed yet. More likely, one or several of the paybacks on the 10s-or-better game are less than what they’d be in an otherwise similar jacks-or-better system. This could easily compensate for the higher hit rate and keep the overall return percentage where the casino wants it. The accompanying pair of tables shows how this might work for two versions of the alternate games, based on play following the optimum strategy for the specific payback schedules.

Configurations of representative 10s-or-better and jacks-or-better video poker machines

10s or Better                                        Jacks or Better			
Hand              Payback   Probability   Return %   Hand             Payback   Probability   Return % 
Royal flush         800      0.0026%      2.0800%    Royal flush       800       0.0025%      2.0000% 
Straight flush      50       0.0104%      0.5200%    Straight flush    50        0.0109%      0.5450% 
Four of a kind      20       0.2357%      4.7140%    Four of a kind    25        0.2363%      5.9075% 
Full house          6        1.1493%      6.8958%    Full house        8         1.1512%      9.2096% 
Flush               5        1.0670%      5.3350%    Flush             6         1.1014%      6.6084% 
Straight            4        1.2386%      4.9544%    Straight          4         1.1233%      4.4932% 
Three of a kind     3        7.4218%      22.2654%   Three of a kind   3         7.4448%      22.3344% 
Two pair            2        12.8936%     25.7872%   Two pair          2         12.9277%     25.8554% 
10s or better       1        25.4368%     25.4368%   Jacks or better   1         21.4583%     21.4583% 
Lose                0        50.5443%     0.0000%    Lose              0         54.5436%     0.0000% 
Overall                      100.0000%    97.9886%   Overall                     100.0000%    98.4118% 

Hit rates are the complements of the probabilities shown for losses. That’s 100 - 50.54 = 49.46 percent for the 10s-or-better and 100 - 54.54 = 45.46 percent for the jacks-or-better examples.

Hit rates notwithstanding, the jacks-or-better version has a superior overall return percentage – 98.41 as opposed to 97.99 percent. Return percentage is more complex than hit rate to ascertain because it’s determined by both the probability of a hand being obtained and the payback.

The contribution to return percentage of each hand is the payback per unit bet – the amount won plus the original wager – multiplied by the probability of its occurrence. Adding the individual return percentages for all hands gives the overall return percentage for the game. The table shows slight offsets between the probabilities for comparable hands in the two games. For instance, the values are 1.0670 and 1.1014 percent for flushes in the 10s-or-better and jacks-or-better machines, respectively. These disparities are caused by changes in optimum strategy to allow for returns on pairs of 10s and payback differences such as 5-for-1 versus 6-for-1 on flushes.

If the 10s-or-better machine had a payback of 25-for-1 and not 20-for-1 on four of a kind, its overall return would be 99.17 percent – exceeding that of the jacks-or-better game. Likewise, leaving the payback for four of a kind at 20-for-1 but raising that on a full house to 7-for-1, the overall return would be 99.14 percent. Conversely, the return on the jacks-or-better game could be trimmed by adjusting its paybacks downward. An illustration might be cutting that for four of a kind to 20-for-1, in which case overall return percentage would drop to 97.23 percent.

Volatility, which influences bankroll swings encountered during a session, also differs for the two configurations, principally because the frequency of low paybacks is greater while that of high returns is less on the 10s-or-better than the jacks-or-better model. The effect is small, however; standard deviations, which can be pictured as representative bankroll jumps per round, are $4.38 and $4.42 per dollar bet on the 10s-or-better and jacks-or-better games, respectively.

For all practical purposes, the two games cited as examples – even when modified by payback changes on selected results – are essentially equivalent from the perspective of the players with respect to net performance during a single session or casino visit. The main point of departure is hit rate, which can have a psychological if not monetary impact. Veteran video poker buffs know that hits on high pairs are really pushes rather than wins. But coins dropping into a tray or registers ratcheting upwards induce powerful illusions which fool most of the people most of the time. And, wasn’t it the renown rhymster, Sumner A Ingmark, who wrote:

Those whose purpose is deceiving,
Know that seeing is believing.

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.