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by Frank Kneeland

Frank Kneeland was the manager of the largest progressive video poker team in Las Vegas, and has authored a book about his adventures entitled, "The Secret World of Video Poker Progressives". You can get the book as well as some extra info about Kneeland on his website In addition, there you'll find the show archive for his new radio show on pro-gambling, "The Winners Edge," and the shows he co-hosted with Bob Dancer for six months, "Gambling with an Edge."

The only thing I've ever been sure about is my overall level of uncertainty, which as it turns out is exactly 8.3% + or 10%. ~FK 2012

This is a continuation of my April BJI article, The Death of Doubt, and the second part of a three-part series on how to prove or disprove the fairness and randomness of video poker machines ... if such things matter to you.

In The Death of Doubt-Part 1, I discussed how I managed to assuage my own personal reservations about the fairness and randomness of video poker machines. If you haven't read that article yet, then I would recommend you do so before reading this one, but it isn't absolutely necessary. Why, you ask? Because, how I did it, isn't necessarily how you are going to do it. Suffice it to say, there was no need to preface that article with a large friendly warning label that read, "Don't Try This at Home," because, how I did it, is quite literally impossible to do at home, or anywhere else for that matter in todayís market (because you'd need to be a manager of a huge progressive team with a few tens of millions of hands as a sample).

Doubt is a personal thing and usually the only person that can remove it is you. If you wonder whether VP machines are random and fair, how could you determine the answer yourself? This is a mathematical question, and the answer to mathematical questions, not too surprisingly, is mathematical.

Chi Squared

No, sorry to say chi squared is not two cups of chi tea. It's a very complicated mathematical formula, specifically designed for determining how closely your results match your expectation. It's the primary tool used to test VP machines by the Nevada Gaming Commission to assure they meet legal standards. Though you won't be able to use raw RNG output as the Commission does, an end user can still apply the same equations on accumulated data. It will still take a "large" number of trials; however, you can achieve a far higher degree of assurance with a far smaller sample using such math, than you could using the brute force looking-at-results-way our team did it, back in the day. With accurate records and the proper application of the chi-squared test, you might not kill your doubt, but you should at least be able to get it hiding behind the couch, under your bed, or in the closet next to the spooky teddy bear that always seems to watch you when you reach for your pajamas.

Naturally, the bigger the sample, the higher your confidence and only you can decide how much confidence is enough ... and by extension, how big a sample you'll need. Anytime we look to a sample as representative of a greater whole, we have to decide on what level of certainty we want. The great mathematician Jakob Bernoulli, famous for his book on probability, Ars Conjectandi, required "moral certainty," which he defined as 99.99% sure. To have that level of certainty about, for instance, election results, you'd have to poll nearly 98% of the population prior to the actual vote! Gosh, might as well just take the vote. In modern times, we have given up on moral certainty in favor of statistical significance ... aka 95% certainty. Sure, we're less sure, but it also takes us less than a lifetime to achieve it. Even though that's another way of saying, 1 out of 20 times you'll be wrong, statistical significance is the modern gold standard used today for all sorts of things, like exit polls and product testing. To reach this level of certainty about video poker machines should be doable in several months to at most a year, depending on how much and how often you play.

If you are concerned right now that I'm about to wax into a long diatribe on chi-squared equations, fear not. If you know them already, you don't need me restating them here. In addition, if you don't know them, you aren't going to want to learn them here. I have incorporated the relevant equations into a spreadsheet, which you will be able to download (details on this will appear in my next article in next months BJI). This month, I'd like to focus on how it works and what the results of the test mean.

Bad Science (or: How NOT to do a Study into Lightning Strikes)

Perhaps one of the hardest things to understand about confidence testing is that you cannot use the test on preexisting data. Imagine for a moment that you wanted to see how likely it was to be struck by lightning. You placed an ad in the paper and asked folks to come in for your survey. The odds are that you would get many people answering the ad, who were invested in the topic (for instance, people who had been struck by lightning). If you based your study on your respondents' past, you'd almost certainly come up with skewed results, nowhere near the true odds of 3 million to one. It would be fine to use the people who answered your ad, as long as you dismissed their past and had them keep records from this point forward.

The same methodology is necessary for testing the confidence of video poker machines. No matter how accurate your previous records might be, you have to start fresh when doing the test. The basic rule of thumb is this: Never use old data, especially if it's what caused you to want to do the test in the first place. The reason for this should be clear. If something unusual happens, such as a bad streak on VP, or getting hit by lightning, it makes people question things, and want to test them; however, without base-rate information, you have no idea how many people out there did the same thing you did and experienced perfectly normal results, never became suspicious, never got electrocuted, and never thought to do a test.

If you have experienced a really unlikely run on VP and then use the chi- squared test to analyze it, the test will tell you it was really unlikely. This should come as no more of a surprise to you than finding out that 100% of the people in the lightning-strike survivorís support-group meeting have indeed been hit by lightning. Don't use old data!

What Chi Will Tell You

The Chi test uses the term, "Null Hypothesis," and it tells you with what level of confidence you can "Reject the Null Hypothesis." This terminology has a tendency to confuse many people, but if we plug it into a simple situation, it should become clear.

You suspect that a friend of yours is using a two-headed coin for flips. He won't let you examine the coin, but he will flip it for you. That the coin is fair becomes the "null hypothesis." If we state that the coin is unfair and two headed, we reject that it is fair, and the only question that remains is how often we'll be wrong.

If your friend flips the coin only once to show you, and it comes up heads, you can boldly state that the coin is unfair, and you'll be wrong 50% of the time ... since perfectly normal two sided coins come up heads all the time (half the time, actually). One heads in a row proves nothing. If he flips it four times, all heads, you can again say the coin is "no bueno," and now you'll only be wrong 6.25% of the time (1 out of 16). To get to statistical significance, you'd need to have him flip it five times, and even then, you'd still be wrong 3.125% of the time. If you were trying to decide on whether or not to ever bet with your "friend" again, that might well be good enough. If he was going to go to prison for fraud based on the results of your test, you might want moral certainty, which would require 10 flips (twice as many), and you still be wrong once in every 1024 times. Keep that in mind before sending anyone to the guillotine. You can never be sure, but you can be "reasonably sure."

In my article next month, I'll detail what to record and how, with a link to download the utility.

DISCLAIMER: I am not trying to prove or disprove that VP machines are fair. Enough people out there in casinos worry about such things that I felt a utility to assist them in testing their concerns and quantifying their confidence would be useful. I'm not asking people to test anything, nor am I suggesting that VP machines need testing.

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