A survey of 4-card hands in video poker

Dec 8, 2009 5:03 PM
Winning Strategies by Elliot Frome |

Last week, I discussed how the human brain can use selective memory to convince a Player that games are not random, when in fact they really are. Rather than rely on selective memory, the question can really be answered by tracking the condition more completely. Rather than remembering the time I discarded a 5 of diamonds, only to draw the 5 of hearts, let’s keep track of every situation where I drew to a 4-card Straight and count the number of times I draw a card of the same rank as the one I discarded.

Doing this by myself would be quite daunting. Besides the number of hours of play it would take to get a decent sample, it would be quite monotonous to sit there and make sure you take notice of every 4-Card Straight and/or 4-Card Flush draw. Fortunately for me, I bumped into John Grochowski at the G2E a couple of weeks ago. For those of you not familiar with John, he is a long-time casino gaming columnist for the Chicago Sun Times and the author of many wonderful books on casino gaming, including The Video Poker Answer Book (available thru Amazon). Fortunately for me, John had already undertaken the process I described.

Twenty-five people, including John, were commissioned to count hands where there was a one-card discard. They played all over the country, including Nevada, California, Missouri, Illinois, Indiana, Michigan and New Jersey. Not every Player tracked everything. Some tracked how often the replacement card was the same denomination/rank AND completion of Straights, some did so only for Flushes and others only for Full Houses. There was no oversight of any of the participants, but none of the participants knew who any of the others were, so it was not possible for any to collude!

There was a total of 78,433 hands observed on which 4-cards were held. Of these, 4,469 resulted in the drawn card being the same rank as the one discarded. This is 5.7%. The theoretical expected result is 6.4% (3 out of 47). While this is a significant discrepancy, the most important thing here is that the occurrence of this was BELOW the expected amount. My best guess would say that the actual amount was probably closer to the theoretical, but the participants missed some of the occurrences. What this does show is that there is hardly some ‘conspiracy’ that would have the casinos replacing the discarded card with the same rank. Of course, this would also bring about the obvious question – WHY, if the game manufacturers were cheating, would they replace the card with the same rank? What would this accomplish? As a programmer, I could just as easily ensure a ‘random’ looking card without using the same rank.

For the 4-card straight draw, there were 32,659 such occurrences of which 5,828 resulted in a straight. This is 17.8% as compared to the expected 17.0%. This time, we’re a bit high, but then again, who is accusing the casinos of causing there to be TOO many straights? When we look at 4-card flush draws, we find that the actual value was 18.8% as compared to 19.1% expected. Again, this is certainly in the ballpark. Finally, with regard to full houses, 7.8% actually became full houses (from a two pair) as compared to the expected 8.5%.

With a total sample of about 80,000 hands, it is not surprising that there would be discrepancies. While we would expect the actual numbers to begin to approach the theoretical values, I’m not sure if this is really enough hands to expect the numbers to be dead on. Also, as John readily admitted, this was not exactly a scientific study. The possibility for human error in amassing the numbers definitely existed. I’m not sure if this study would pass FDA muster. I’m sure that the naysayer who swears that the machines are rigged will point to this as a fatal flaw in the study. Instead they’ll try and convince you that they’ve seen cases where same-rank replacement regularly occurs 20-30% of the time.

Rather than argue and hurl insults (John told me he got some rather nasty e-mails after he published his results), I would suggest that perhaps both sides get together and try and get a more complete and perhaps scientific study going to put this issue to bed once and for all! I, personally, have very little doubt what the results will say in the end.

You can try out your strategy by playing our video poker game.

Question? Comment? E-mail me at: Elliot Frome