Video poker same rank draw experiment

One of the first things I learned about video poker a couple of decades ago is that results are as random as humans can program a computer to be. There is no such thing as perfect randomness, but programmers can get close enough that we might as well say that every card has an equal chance of turning up on every deal and every draw.

That's what I've been telling everyone who reads my column or who has heard me speak for going on 16 years now. It squares with what programmers have told me, what other expert players have told me, and with my own experiences.

A few months ago, however, I was told of a study that challenges such assumptions. A player had made nearly 5,000 one-card draws, and had received a card of the same denomination as his discard 47% of the time. For example, dealt 9s of hearts and spades, 7s of clubs and diamonds, and a 3 of spades, he would discard the 3, and receive another 3 in its place nearly half the time. In a perfectly random game in which every card has an equal chance of being dealt, that should happen only 3 times per 47 trials, or 6.4% of the time.

I've never seen results anything like that on a video poker game. In fact, over the years, when people have told me they were seeing more same-card draws than they thought they should, I've told them the effect is mostly selective memory, and that if they keep accurate track they'll find the effect disappears. No one has ever written back to say that careful count was still finding a larger-than-normal percentage of same-card draws.

So I decided to do my own large-scale test. I enlisted help from both the VPFree video poker site (http://groups.yahoo.com/group/vpFREE/) and from video poker author Linda Boyd's forum at the Midwest Gaming and Travel Magazine site (www.midwestgamingandtravel.com). Along with some friends and video-poker playing acquaintances I coerced into helping, I wound up with 24 people trying to track the following data as they played:

• Total number of one-card draws and number of times the replacement card is same suit, different denomination than the one-card draw.
• Number of hands holding four parts of a flush, and number of flushes completed with the one-card draw.
• Number of hands holding four parts of an open-ended straight, and number of straights drawn.
• Number of hands holding two pairs, and number of full houses drawn.

In a trial lasting just over two months from late April through the end of June, I hoped to get a very large sample, perhaps 100,000 one-card draws. As it turned out, I would up a little short of that, at 87,379 hands. There was a flurry of activity at the beginning of the trial, but some player quickly tired of the work involved in tracking data.

Only three of the players tracked every category. Most opted to track one or two categories — just same-denomination draws, or just flushes and straights. My data is skimpiest on full houses, where only six players tracked that aspect.

I make no claims on this being a scientific study. Every player was on his or her own, with no observation. However, I took care to avoid any comparing of notes. Other players were not told who else was participating. Indeed, most do not know each other. Play was spread through several states — Nevada, California, Missouri, Illinois, Indiana, Michigan and New Jersey.

With those cautions, here are the results:

• Same denomination replacements: 78,433 trials; 4,469 same denomination, 5.7%. This is a little lower than the 6.4% expectation listed above. That's likely partly due to chance, and partly due to situations such as jack-9-6-3 of hearts, with a 9 of clubs. Expert play is to discard the 9 of clubs, but there are only two more cards of the same denomination in the deck instead of the usual three.
• Flush completion: 39,788 draws, 749 flushes completed, 18.8%, slightly less than the expected 19.1%.
• Straight completion: 32,659 draws, 582 straights completed, 17.8%, somewhat more than the expected 17.0%.
• Full house completion: 4,231 draws, 329 full houses completed, 7.8%, somewhat less than the expected 8.5%.

Real-world results are never going to come up perfectly spot on mathematical expectation, but the players in this trial reported results that are very close to what we'd expect to see in a random game.

One final note. Over the course of this study, I was accused of bias both by people who thought I would steer results to support the study that found a greater percentage of same-denomination draws, and by others who thought I would steer results to debunk it. Neither was the case. I was just trying to gather data, and using other players seemed like the fastest way to get into the realm of large numbers.

I do not claim to have debunked anyone else's work. I just tell you that my group's results found no large deviance from the expected norm.