Today’s paper is on simple video poker mathematics. Let’s assume you are playing a game where, on average, you hit a quad (i.e., a 4-of-a-kind) every 400 hands. Further, let’s assume you play for a total of 1,200 hands. I’ll arbitrarily say that it takes you two hours to complete the 1,200 hands. How many quads can you expect to end up with over that number of hands?
It appears obvious that the answer should be three, but this is the wrong answer. To get the correct answer, we need to look at the binomial distribution, the results of which appear here:
| 0 | 5% |
| 1 | 15% |
| 2 | 22% |
| 3 | 22% |
| 4 | 17% |
| 5 | 10% |
| 6 | 5% |
| 7 | 2% |
| 8 or more | 1% |
What this says is that 5% of the time you won’t hit any quad; 17% of the time you’ll hit four; 2% of the time you’ll hit seven; etc. These numbers don’t tell you WHICH quad you’ll hit. Just how many.
These numbers are accurate, but not really precise. For example, the chance to get exactly three quads could more precisely be written as 22.4322%, but that is far more precision than we need for today’s discussion. It looks like they only add up to 99%, but that’s rounding error and also not important for today.
One of the interesting features of this distribution is that the number of quads that we think we “should” get, namely three, actually occurs less than one time in four. Another typical feature of the distribution is that the probability of getting one fewer quad than typical is virtually the same — actually 22.4135%, which is slightly less.
We could, I suppose, refer to getting either zero or one quad as “bad luck”, getting two, three, or four as “typical luck”, and getting five or more as “good luck”. It doesn’t change anything by assigning terms dealing with luck to the results. When somebody asks me, “How much skill and how much luck was involved?” in describing whatever happened yesterday, my answer is often, “I have no idea.”
Let’s assume that on this particular day in question, we don’t hit any 4-of-a-kind. Definitely worse-than-average luck, but it happens about one day in twenty. Slightly rare, but not extraordinarily so. Now the question is, since you’ve just gone through worse-than-average luck, what will be the distribution of quads for your two-hour session tomorrow? For this, the following distribution will hold:
| 0 | 5% |
| 1 | 15% |
| 2 | 22% |
| 3 | 22% |
| 4 | 17% |
| 5 | 10% |
| 6 | 5% |
| 7 | 2% |
| 8 or more | 1% |
The distribution, of course, is the same as first given. Just because we had a bad day says absolutely nothing about what our score will be the next day. There is no tendency to either, “Once you start running bad you keep running bad because you’re an unlucky player,” or “You’ll get more quads the next day to make up for the shortfall.”
Let’s assume we change machines halfway through. Now the distribution of the quads expected over the 1,200 hands is:
| 0 | 5% |
| 1 | 15% |
| 2 | 22% |
| 3 | 22% |
| 4 | 17% |
| 5 | 10% |
| 6 | 5% |
| 7 | 2% |
| 8 or more | 1% |
Is this distribution beginning to look familiar? It should. Changing machines has nothing to do with changing the distribution.
In this discussion so far, we’ve said nothing about skill. We are assuming players are playing perfectly. If players play imperfectly, the distribution will change. For example, on a hand like K♥ K♠ 4♦ 4♣ 5♦, it is correct in almost every game to hold KK44, although many seat-of-the-pants players playing games where two pair only return even money incorrectly hold just the pair of kings. Making this kind of mistake systematically will IMPROVE your chances for hitting quads, but COST you overall. The increased number of quads you get by holding only one pair rarely compensates for the reduced number of full houses.
The numbers are for three “cycles.” If full houses normally come around every 90 hands on average, the numbers above apply to how many full houses you hit in 270 hands. If royals come about every 40,000 hands, the numbers above apply to how many royals you hit in 120,000 hands. In games where the royal cycle is 45,000 hands, the numbers apply to how many royals you hit in 135,000 hands.
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So does it change for multi line for the percentages such as 3 line or 100 line?
I played about a 1 hour session of 9/6 JoB Tuesday afternoon. I got 4 quads, which as shown, is nothing unusual. The first three though, were all JJJJx by holding JJ alone! The last was QQQQA holding QQQ.
Yes it changes. Hundred Play is NOT the same as 100 independent trials, because each hand starts from the same initial hold.
It’s an interesting question and worthy of a longer discussion. I’ll discuss this soon — but not today.
A different way to look at a cycle of play is that you are equally likely to hit 0 or 1 of the event and almost as likely to hit 2 or more of the event.
Your chance of hitting 0 quads in 400 hands is 36.74%. Your chance of hitting 1 quad is 36.84%. Your chance of hitting 2 or more is 26.42%.
It’s interesting that hitting the ‘expected’ number of quads in a cycle is about the same as hitting 0 quads in that cycle.
I have experienced so many short-term bizarre results that I’ve questioned the randomness of the machines–as those anomalous results have always been negative. I always get shouted down when I mention those results, though, by people who assume that I’m overweighting small sample sizes (“I didn’t hit a single quad today”). The reactions have been particularly vociferous when I mention such results on the website of someone who makes their living on making people believe that video poker is 100% honest. Yet, am I just the unluckiest guy on the planet when I, in less than a year’s time: 1) play 6000 hands of DB without hitting a quad; 2) go 0-for-51 on one-card flush draws on JOB; 3) miss 127 one-card royal draws, hit one, and then miss the next 120 4) play 50,000 hands (various games) without hitting a royal? And yes, yes, go ahead and give me the usual sarcastic reply, accuse me of selective memory or data bias, whatever you wish, but I do keep careful records. I’ve had too many billion-to-one-against occurrences. Here’s the $64,000 question: at what point should we become suspicious? It isn’t necessarily random fluctuation any more when you get hit by lightning twelve times in a week. At SOME point–we all have our thresholds–we are forced to use Occam’s Razor: 1) I’m the unluckiest person in the history of the universe, or 2) I’m being cheated. Which does Occam’s Razor suggest?
Kevin writes a lot of food for thought. Is there a good gamble left in Las Vegas?
If anybody suspected video poker was not random, they would not be playing.
Nobody plays blackjack anymore because a blackjack only pays 6 to 5.
That leaves the interesting idea of dice control (written about in a couple of books), but since dice tables now are 14 to 16 feet long, I kind of write that one off as well.
Really . . . . . . . the most fun I have gambling in Las Vegas? At breakfast in the cafe, I buy a $2 keno ticket and watch the numbers come up. The people I’m with laugh at me, but if I get 2 or 3 numbers, they all start hollering and laughing and carrying on!
Maybe someday I’ll buy and read a couple of books about sports betting. Maybe there is something there I’ve missed.
I wish you were right about people avoiding 6:5 blackjack and other crappy games. The casinos actually were very tentative about introducing 6:5 blackjack, reasoning that only a fool would tolerate getting paid less for a natural, the best hand in the game. In fact, at first they introduced single deck 6:5, touting the lie that single deck (rather than a shoe) made up for the reduced BJ payout. Then they discovered (again, gradually) that they could deal 6:5 shoe games with no loss of business. People just didn’t care that the house advantage had been increased by 800%. In fact, some people, being even more math-challenged than average, thought that a 6:5 game is BETTER than a 3:2 game because, y’know, 6 is bigger than 3!!!
Isn’t this similar to a poisson distribution?
“I have experienced so many short-term bizarre results that I’ve questioned the randomness of the machines–as those anomalous results have always been negative”
I won’t shout you down. I have seen some really ODD sh*t playing blackjack where it is controlled by “video lottery” vs. playing live hands. To the order of most of my losses came from those rather than from actual table games.
If you look at the laws governing certain class II games (including those made to look like class III games), you’ll see that these machines are doing things that no table version of the game can do. For example: constantly running “randomness routines” in the background while the game is played. Umm that’s like saying the dealer is shuffling the deck all the time, even during play. That doesn’t happen.
But it’s all legal because the law says that the machines are allowed to do this AND if they do anything else “anomalous” then so long as it’s stated somewhere on the help screen (usually in itty bitty type), then it’s legal (even if it biases the game further in favor of the house).
Thank you sondjata for your post. That was some interesting information.
another way to look at this is if three gamblers each play a royal cycle of jacks or better, on average, one gambler will hit no royals and lose on average even more than the royal is worth, the second gambler will hit one royal and be bored, since on average they will still be a loser, while the third gambler will hit two or more royals and quit their day job to become a “professional gambler” (as long as their lucky streak holds out)
Ya know, when I’m not hitting quads, I switch to DW. Then the naturals come out of the woodwork! 😉
While over any 1200-hand stretch, those percentages are true, isn’t it also true that over the long run, the expected number of quads in a 1200-hand stretch is, in fact, 3?
Well, yeah, sort of.
Let’s say you played 1,200,000 hands (which is 3,000 400-hand “cycles”), you will get “approximately” 3,000 quads — but there will be a range between about 2,800 and 3,200 where you will actually end up. Hitting 3,000 on the nose is very unlikely — but not impossible. It is unlikely, but not impossible, that you could end up with less than 2,800 quads or more than 3,200 quads.
It’s always going to be a distribution — like a “bell shaped curve.”
If you have played 1,199,900 hands (i.e. just 100 less than the 1,200,000 target) and you’ve had 2,950 quads so far, you cannot depend on getting 50 quads in 100 hands. (That would be almost impossible.) There is no “video poker scorekeeper in the sky” keeping track of all of this to make sure you get your due.
Also, skill comes into play here. If you play hands incorrectly according to perfect strategy, the number of quads will go up or down.
Isn’t this similar to a poisson distribution?
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The Poisson Distribution and Binomial Distribution are similar — but not identical.
I do not want to get into a seminar on statistics here. If you Google “difference between Poisson and Binomial Distributions” you’ll end up with a complete discussion — which is heavy on the math. Go for it, if you like
isn’t that the truth
I too have been questioning the randomness of the video poker machines in the casinos I play in Las Vegas for some time now. I started out learning to play video poker in 2004 with help of “WinPoker”. When “Video Poker for Winners” came out, I bought the program, and I use it all the time to improve my skills. I also keep meticulous records because I play every day somewhere depending on the mailers I receive. Just this past week I played NSU for a couple of hours every day in four different casinos. I can forget getting four deuces. In just over 7000 hands, I didn’t get four deuces once. I got three deuces twice, and I’m talking about on the deal and after the draw. You can say, “oh it’s a small sample”, but this has been normal rather than the exception for a long time. Three weeks ago I played every day, and I made a note every time I got any three of a kind. At the end of the week, it was interesting to see that every card in the deck received almost exactly the same number of three of a kinds plus a few quads, but not the deuces. When I can’t even catch three deuces, it makes you wonder where have the other two deuces gone in randomness of NSU.
7000 hands is about 1.5 cycles. You’ll go through a LOT longer dry spells than that.
I understand you’re unhappy about it, but that’s hardly persuasive evidence that the games are unfair
When at home and bored, I’ll deal out 200 hand “sessions” of 9/6 JoB based on a 25 cent game and have kept records of just amount won and lost. This is an on going project I’ve been doing for a while. I don’t know the number of quads, but there have been plenty of times I’d deal out 6 or more “sessions” without a quad. And to add to that, I’m near 30,000 total hands and have only 1 straight flush. No royal yet.
Naturally, you’ll have the gripers and conspiracy theorists on various forums complaining that the games are rigged.
Just today, a lady next to me was playing Extra Draw Frenzy TDB. She was dealt 3 of a kind four times. 2’s twice, 3’s once and kings once. Getting dealt trips (No full house) initiates the bonus frenzy hands. She had a 24, 24, 26, 26 hands respectively for a total of 100 frenzy hands. Now mind you, in the frenzy bonus, you cannot hold a kicker with the trip 2’s, 3’s, 4’s and A’s as optimal strategy would call for in TDB. She hit only one quad (kings) out of the 100 hands.
I play DDB almost exclusively, with perfect strategy, how often will you see a 4OAK?
About the same as Jacks or Better (which is 1 in 423.xx for the 9/6 version. I used 400 in the article because it was easier to work with.) The actual number varies slightly by game and pay schedule and what strategy you use. In DDB, and other games, the strategy differs depending on the pay schedule — but many players use the same strategy for all pay schedules and think they are playing perfectly.
I am a recreational player in Atlantic City and first time responder. Hello to all…Bob I have greatly admired your seeming single mindedness concerning long term winning. Started reading your articles in Strictly Slots ten years or so ago still find useful information in “Million Dollar Video Poker”. However, I’m confused as to how one can apply a statistical analysis to a situation where a casino’s bottom line may dictate a particular hold pattern on its slot machines. Video Poker, at least in Atlantic City , is random but not “fair”, let the buyer be ware…l have practiced on the earlier version of “Win Poker” and”Video Poker Wizard” and never had the high hit frequency I had as in those practice seasons. Recently however, I lucked into two Royals about twenty minutes apart at the Borgata..no complaints. Thanks for being out there and doing what you do.
You can’t really win money unless you bet everything at once during the first play and win.
[Editor’s Note: This quote from an online-gambling portal called Casino Guru.] “These games are somehow set to give casinos a certain advantage in the long-run. Otherwise, the owners of the casino would be losing money by running those games.”
You are welcome to your opinion — but I strongly disagree with your premise
There are several casino games where skill matters — and sometimes casinos inadvertently-or-otherwise leave holes where knowledgeable gamblers have the advantage. Even in games where skill does not matter, it’s possible that casinos miscalculate the benefits they are awarding for playing the game. That too, is a “hole.”
I “live” in those holes. That’s what “Gambling with an Edge” is all about.
Your post has just provided me a topic for a future column. Look for it in early April. Thank you!
Your grasp of statistical math is quite impressive Bob d. !!