This column is inspired by an email I received about 20 years ago, when TITO tickets weren’t found everywhere. I don’t have the email in front of me, but I remember the gist of it and certainly the line that I’m using as a title. For the rest, I’m using some artistic license that I believe is fairly close to the original.
Dear Mr. Dancer:
Life is so unfair!
I was playing quarter 9/6 Jacks or Better at my local casino. With the 0.67% cash slot club, it’s mildly positive. The lady next to me, let’s call her “Lucky Lucy,” was playing 9/5 Double Double Bonus, a game that you say is so bad that it should be avoided.
Anyway, Lucky Lucy was dealt AAAA2 for a $500 hand pay. Three hands later LL got 22223. The 800 quarters started to spill into her tray, but the hopper went dry before she got the whole $200. So, they came and filled up the machine. About 10 hands later, LL nailed a $1,000 royal flush!
I’m playing the so-called good game and losing my ass! In less than 20 hands, which took more than 20 minutes because she needed so much servicing from the casino staff, on a terrible game, she was ahead $1,700. I’m starting to believe that pay schedules mean squat. You’re either lucky or you’re not!
And random – smandom! If you think this was a random result you can bite me!
Frustrated Fred
Dear FF:
Yes, I think it was a random result, but I respectfully decline your culinary invitation.
LL had an extremely lucky run that she’ll be talking about for the rest of her life! Thirty years from now, she’ll be saying, “Let me tell you about that time back in 1996 when . . . .“ She was playing a less-than-98% game that normally eats her lunch. There will be ugly stretches where that game pays less than 90% over a few thousand hands — as well as very occasional times where she wins big. Even including the “never-in-my-wildest-dreams” session you just described, she’ll be a big loser on this game over time.
Could it happen that she quits forever and ends up a net winner on that terrible game after such a wonderful run? Theoretically, I suppose, but it usually doesn’t work that way. What is more likely is that she’ll come back as soon as possible to see if she can capture lightning in a bottle one more time. And the most likely result is that she is going to lose — because that’s the nature of that game.
You, on the other hand, are playing a dull little game where, over time, you’re going to lose almost a half percent, which is more than offset by the generous slot club. There will be days you win and more days you lose, but over time it will come pretty close to the half percent it’s supposed to (assuming you play well) and you’ll be a net winner after collecting your slot club benefits.
What “random” means in this case is that the results mimic those of a freshly shuffled fair deck. Sometimes you’re randomly dealt four aces and a kicker (one time in 216,580, if you’re counting). Unusual? Yes. It certainly doesn’t happen every day or even every year to a given player. But it happens. It happened to LL while you were sitting next to her and she got $500. It’ll happen to you just as often, maybe next time will be a year or three down the road, but you’ll only get $31.25. At times like that, it’s hard to see that receiving an extra $1.25 for every time you end up with two pair pays off better in the long run. But it does! If you want to complain about playing the wrong game when such a nice hand is dealt, you won’t be the first one to do so! Many video poker players complain a lot!
“Random” includes lots of results that are surprising because they happened TODAY. It never happens that you play 1,000 hands and get the exactly predicted number of every hand. It can’t happen because some hands have cycles much longer than 1,000 hands. It’s going to take about 40 of those 1,000-hand cycles to receive a royal, and 650 of those cycles to be dealt a royal. But dealt royals happen randomly — once every 649,740 hands on average. Keep playing and you’ll occasionally be dealt a royal. It happens. Randomly!
Good video poker players have come to believe that over time, the results end up where they should. Over millions of hands, you’ll end up with approximately the correct number of royals, straight flushes, 3-of-a-kinds, etc. If you’re playing where you have the advantage, you’re very likely to be ahead after millions of hands. If you’re behind after millions of hands, it most likely is because you were playing games where you did not have the advantage.
Can I guarantee this? No. Of course not. Depending on how big your edge is and how many hands you’ve played, you might be an 80% favorite to be ahead, or a 90% favorite, or a 99.993% favorite, or whatever. You will never be a 100% favorite to be ahead, but we don’t live our lives with 100% guarantees. (You can’t 100% guarantee that you’ll be alive a week from now, for example.)
But you can BET you’ll be ahead, and really that’s what we’re doing when we gamble. It can be a very, very smart bet to make, even if we can’t be positive that we’ll always win.
But even though I can’t guarantee I’ll win over the next however-many years, I 100% believe I’ll do very well and am betting considerable amounts that it’s true.
Great article ! We all know people that understand the math (or at least accept it) and we know others that simply do not get it. One person that I used to visit casinos with was playing 5-8 something. When I pointed out that the same game was in the next bank for 6-9 he said that this was his lucky bank of machines and he would not move to the better machines. That evening at dinner I thanked him for his playing choices because if we all played the best machines with perfect strategy the casino would be more likely to tighten things up. We did not travel together much after that.
It’s a fallacy, however, that things tend to “even out over time.” They do not. The actuality is that (somewhat paradoxically) results tend to approach expectation over time but the difference between results and expectation becomes larger. So let’s take my own experiences as an example. One horrible three-month stretch at.25 FPDW: 300,000 hands with only one royal, and about a dozen fewer 2222 than expectation. Minus $7,200. Now, am I ever going to get that back? Are my lifetime results ever going to reflect the 0.76% advantage I had? No. The reality is that i will NEVER get even for my lifetime FPDW play, just because I’ve had several terrible stretches like that and no corresponding big winning streaks–the best was earlier this year when I was four royals ahead for Jan-March, for about +$3500.
Conversely, there are plenty of people who play garbage machines, or good machines badly, but have gotten so lucky that they’ll probably NEVER wind up in the red. I remember Crazy Mary. She lived at the Fiesta (figuratively, but maybe literally as well) back when they had oodles of FPDW. She would arrive at her favorite bank of machines, and if every one was taken, she’d stare at the players until one, unnerved, left. She would then play until she hit the deuces or a royal. Her strategy was terrible. She would make plays like holding A10 suited. And she won. And won. And won. She would actually get very peeved when two or three hours went by and she didn’t get “her” 2222 or royal. She eventually used her .25 FPDW winnings to buy a car. And promptly wrecked it. But that’s another story.
My point is that SHEER, DUMB, RANDOM good or bad luck has MUCH more to do with your results that anything else until you get into the very, very long term. That, of course, doesn’t mean that you shoudn’t play the best machines and use the best strategy. But yeah, empathizing with Mr. Bite Me, I know that you can study strategy charts until your head bursts and sit there pounding the buttons and inhaling the smoke, and you’ll be down $5000 when some LOL shambles up to the 6/5 Bonus Poker machine next to you, sticks in a $5 bill, and gets dealt a royal.
ALWAYS, Just one hand away from greatness ,or, conversely Life is a bitch and then you die. your choice 🙂
VP/ gambling is just one long game. Fun to play, maybe not so much fun if it is to pay the rent.
Have a Big enough bank roll zero ROR .
You do have to get into the very very long term. And this can mean one person’s video poker play over their lifetime. I personally estimate that if I wanted to get one machine into long term average return, I would play it pretty regularly for about three years.
Long long term is the hardest part of being an advantage player.
“Give me some of the yellow and don’t be cheap on me.”
I want to carry on with Kevin’s and Jeffrey’s comments, by mentioning a couple of things even they didn’t bring up. You have the greatest chance of being at or finishing at a game’s EV (e.g., ahead by 0.76% when playing FPDW) if you play on the same one machine for all your play, and if nobody else plays on that machine. But that never happens. Most of us play on more than one machine from trip to trip. But all of the machines on which we play are played by other gamblers between our trips. So it will be very difficult for us to land right on the EV. There’s your automatic deviance from the expected EV right away. But now bring into the picture the fact that players’ results will not all be the same, but will vary like a modified bell curve. Those at the bottom “lip” of the graphed results will most likely never be able to get back up to break-even no matter how many millions of hands they play. And the same is true for those at the top lip, in the reverse way. The only thing we can count on with high probability is that the total combined results of all players on a certain game will be near the EV (minus the percent of inferior hand-playing choices) in the very long run. But we cannot guarantee that any particular player will finish at or even near the game’s EV.
Outstanding column! It is the general feeling that most casino hobbyists have, a good streak of luck that will bring them back to the casino for more. One of these moments is enough for these players to neglect losing in which Mr. Dancer is well aware of the good times as well as the ugly times.
You have the greatest chance of being at or finishing at a game’s EV (e.g., ahead by 0.76% when playing FPDW) if you play on the same one machine for all your play, and if nobody else plays on that machine.
————————
Absolutely not!
Playing on one machine or 50 has no bearing on your short term or long term EV
Being the sole player on a machine or being one of many has no bearing on your short term or long term EV
A lot of people do believe that a VP machine has only so many “hits” in it and when they’re gone, the machine is a dried husk. Or conversely, one that hasn’t paid off in a long time is obviously bursting with imminent wins, like a pregnant hippo.
There’s one little old Oriental guy who haunts the fullpay banks at a locals’ casino in Vegas. He sticks a $5 bill in one machine and plays for five minutes. Then he cashes out and moves to the next machine. Always counterclockwise, stopping to glare and mutter curses at anyone who is playing the next machine in the sequence. When playing, he hits the draw button and then throws his hand in the air, like a deranged symphony conductor. When he runs out of money, he goes home (and his strategy is pretty bad). I’m sure he’s convinced that his precise methods are making him win. And at this point, he’s so invested in the ritual–self-brainwashed, if you like–that not following it would badly traumatize him.
The human brain has a very, very hard time with the concepts of probability and randomness. So much of the science of probability and statistics is counterintuitive. We want–desperately–to find patterns in things that are random. Assuming honest machines, playing one hand on each of five machines is exactly the same as playing five hands on any one of those machines. It’s all random, no matter how much we try to alter or deny that randomness.
I do not get to town much but is that the guy a Boulder Station?
To elaborate on the above comment, i have always contended that money brings out the worse traits in people. There are characters in the local casino that make it a habit to put exaggerated arm and hand motions into their play. This is in addition to the normal swearing, grunts, and cries of pain when the draw does not give them the desired results. Players are always looking for sympathy or give you that deer in the head lights look as you pass by.
It’s all about cycles.
Play 100 cycles and you can expect a 3SD distribution of +/- 30, in other words somewhere between 70 to 130 hits. If those are royals, most royals pay 800 bets, so that would be somewhere between 56,000 to 104,000 bets, a pretty substantial range of possibilities.
Up your game and play 1000 cycles? 3SD would be +/- 95. How about 10,000 cycles then? 3SD would be +/- 300. See any patterns yet? The more you play, the less likely it is that you will have an average result.
BTW, your particular royal cycle depends on how you play, and can vary widely, but let’s take a typical 40,000 hands. 10,000 cycles would be 400 million hands. Good luck with that one! And you think that might qualify as “long long term”?
As a side note, I wonder if Bob could estimate how many hands of video poker he has played? It’s hard to estimate, but if I were to guess I would guess he’s played maybe something like 100 million hands.
According to the strategy maker for 9/6 JoB on Wizardofodds.com, there are 1,661,102,543,100 different possible outcomes with optimal strategy. That’s 1.66 TRILLION!! 100 MILLION still isn’t close to the “long run.”
Every one of us has seen stretches of “unusual” outcomes, from missing 4 to a royals over hundreds of times to seeing the king of spades pop up 6 straight times while holding 4 card combos. Stuff happens.
Wow… there is a lot of not understanding the basics of the game going on here…
“You have the greatest chance of being at or finishing at a game’s EV (e.g., ahead by 0.76% when playing FPDW) if you play on the same one machine for all your play, and if nobody else plays on that machine.”
It’s crazy that people can actually think this. Even a minimum amount of critical thinking would show you that this makes no sense. For this to be true, a machine would have to not only ‘remember’ all of the hands it has given you, but it also has to know that it was you playing the whole time.
If your contention is that you can only get your long-term EV by playing on one machine, what if you and a clone of yourself took turns playing one machine? Would you still get that long-term EV? Yes, right – I mean, you’re both basically playing as one person… so then what about if instead of a clone it was a complete stranger? If you think that changes things, how does that make any sense?
Well, when I insert my slot card into the reader, it says, “Hello, Stupid” or “Crap, it’s YOU again,” so maybe the machine DOES know it’s me. Certainly, that would account for its proclivity to torture me by when I draw to the AKQJ of spades, giving me the 10 of clubs. It’s hoping I’ll have a heart attack.
I love what LC Larry wrote. I remember a lot of odd sequences that just stick in your memory. Once I was playing 9/6 jacks or better at Main Street Station and I was dealt a pair of fives twelve hands in a row. That was weird, but actually it should be expected from time to time (over the long run).
Video poker involves the “skill of the draw” so although you can never know your true EV in the casino, and I guarantee it’s not the computer perfect EV number, it’s important to make an educated guess and to use Bayesian to fine tune your guess. Even though you will never play enough to “get the EV”, your true EV factors into your trip bankrolls, your approximate Kelly bankroll (variance/EV), and your Nzero (variance/EV^2), and all those are very important if you want to be a winning gambler.
Liz, would you agree that if a given video poker hand (such as a natural royal flush), is below its expected observance rate (assuming optimal discarding) that, at some point, there has to be a stretch of hands where the hand must be above its expected observance in order to converge on its theoretical appearance rate ?
According to the law of large numbers a hand must eventually converge to its theoretical probability of appearance and this can’t happen if an under-represented hand doesn’t appear above its average over some sequence of deal-draws.
Jumping the gun before Liz replies —-
There is no “video poker scorekeeper in the sky’ to make sure that you’ve been given the requisite number of any particular types of hands. However many million hands you play in your career may well end up under- or over- royaled.
Will you have under-royaled intervals? Sure.
Will you have over-royaled intervals? Sure.
Will there be the same number of each? Possible, but doubtful. It actually difficult to determine when each interval starts and stops.
Will some people whine more during their bad stretches? Sure.
Will some players overestimate the level of skill during their over-royaled stretches? Absolutely.
You should always assume that going-forward, you’re going to have average results (given the game, slot club, skill level). There will be variance. If a couple of standard deviations down will bankrupt you, you can’t afford the game.
There are players who will bet more during winning streaks figuring their good fortune will continue. there are players who will bet more during losing streaks figuring the law of large numbers will save them in the end.
Both are wrong.
So the law of large numbers applies to Casino house edges (which allows the casino to make money on its -EV games) since the game converges to its house edge, but ceases to exist when it applies to the convergence in expectation of an under-represented video poker hand (which doesn’t have to be a royal, BTW, the hand could be an AWAK, or some other premium hand with a higher probability of occurrence than a Royal Flush). The law of large numbers is amazingly discerning, knowing exactly when to apply itself and all (to the casino and not the player).
The law of large numbers is greatly misunderstood. Yes, results do converge to theoretical expectation over time, but the AMOUNT of the divergence grows over time as well. Say you play 100,000 hands of FPDW and you are 0.7% under expectation. Later, you have accumulated 1,000,000 hands and you have narrowed it to 0.2% under. Your results are closer to theoretical but the divergence is larger. The law of large numbers says that you can expect that sort of thing.
If you consider a hypothetical population of APs, playing a +1.0% game perfectly, after they all play 1,000,000 hands, some of them will be down by 1% or more (the exact number depending on the game’s volatility). This is another manifestation of the law or large numbers: as the sample size increases, results converge to expectation but the number of anomalous results increases.
I remember studying the law of uneven distribution in college. Results of this game and many other probability games are non-linear. There is a much greater chance of hitting those big payouts in a small sample size of dealt hands than there is of hitting one everyday. As Bob says, some days you’ll hit nothing, some days you’ll have the miracle machine. It seems like its the miracle machine but its a result or a cluster of random results. The person in the example given in the letter will get destroyed over time playing against those odds.
Kevin, thanks for the reply. Is there a good source of reading material you can point me to (for example a hyperlink) with regard to the mathematical/statistical definition of divergence within the context of the Law of Large Numbers ?
Right now I am still wrestling with the idea that a casino can take advantage of an event X’s probability of occurring (house edge of a casino game) and yet the player cannot take advantage of the probability of event Y occurring (an AWAK for example) even though both are events (apparently nature cares what the event is) with associated probabilities and both must converge to their expected probability (eventually).
It’s the edge that makes the difference. If you play a breakeven game, the eventual result is bankruptcy, not breakeven, this is because the negative swings increase in value until eventually your bankroll (your ability to absorb negative downswings) is exhausted. Now, if there is an edge, Nzero takes over. Nzero is variance/edge^2 hands and is the point at which the edge catches up to 1SD. At that point the player with the edge has an 84% chance of being ahead and a 16% chance of being behind. For less than Nzero hands, the chances aren’t as good, for more than Nzero hands, the chances of winning are better.
As for “regression to the mean” (see wikipedia for terminology), as long as the shuffle is truly random, there is no regression to the mean. If the shuffle has flaws, then there can be a clumping effect where the next hand is effected by the previous hands. This is called “shuffle tracking”. PRNG’s can also display these kinds of flaws with similar effects.
If you want a pretty good, albeit technical, explanation of convergence, check out the first part of this article:
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3095954/ or this article:
https://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/
The best explanation of the paradox of convergence/divergence was in my textbook for my statistics and probability class at U of Oregon, which I freely admit I took just to fill in a time slot. I don’t remember the name of the textbook. I did get an A in the class though 🙂
Briefly, the “paradox” described therein is that as the sample size grows larger, results converge on expected value BUT the likelihood of those results landing EXACTLY on EV approaches zero; furthermore, as sample size grows, the amount of the divergence increases on an absolute basis even as it decreases on a percentage basis.
The answer to your speculation is pretty much that there is no way to “take advantage” of negative expected value. The fact that the player experiences negative EV only in the short term is what allows him to sometimes win. If every time you visited the casino, you played craps for a million rolls, you would pretty much NEVER win.
And BTW, Liz is incorrect about regression to the mean. Random events DO regress to the mean. In fact, a fair shuffle guarantees it, and a flawed shuffle keeps it from happening. Also, positive EV does not insulate one against bankruptcy. In fact, only an infinite bankroll can survive infinite iterations of the game, even with a 99% player advantage (a 100% player advantage would do it, though physical survival might be an issue, as in RoER (Risk of Emergency Room)).
“And BTW, Liz is incorrect about regression to the mean. Random events DO regress to the mean”
If there is regression to the mean in video poker, then you should definitely increase your bet size if you are losing and decrease your bet size if you are winning. It’s the equivalent of “buy low, sell high” in the equity market which does exhibit regression to the mean during times of stability.
“If there is regression to the mean in video poker, then you should definitely increase your bet size if you are losing and decrease your bet size if you are winning. It’s the equivalent of “buy low, sell high” in the equity market which does exhibit regression to the mean during times of stability”
Liz, this technique is so popular in the stock market, that it actually has a name – Dollar Cost Averaging (DCA) :
https://www.investopedia.com/terms/d/dollarcostaveraging.asp
Thanks Liz. Given the statement “As for “regression to the mean” (see wikipedia for terminology), as long as the shuffle is truly random, there is no regression to the mean.”, it’s amazing to me that insurance companies and casinos, both of which rely on this flawed concept of regression to the mean, are able to make any money since there is no guarantee of convergence to the house edge (or the probability of a catastrophic event in the case of insurance companies).
At any rate, I am going to investigate a concept called Large Deviation Theory (I got the idea to do this from Kevin Lewis’ reply about divergence) which seems to have a well-developed mathematical formulation, along with its relationship to the law of large numbers as well as the paper:
“Relation between the rate of convergence of strong law of large numbers and the rate of concentration of Bayesian prior in game-theoretic probability” which has the link https://arxiv.org/pdf/1604.07911 .
Before you had explained that there is no regression to the mean in a truly random process, I would have still been mystified as to why Mother Nature chooses to enforce the law of large numbers when it comes to the house edge of a casino game but doesn’t choose to do so when it comes to the probability of getting a premium hand in a given video poker game.
Thanks again to you Liz and to Kevin Lewis.
“it’s amazing to me that insurance companies and casinos, both of which rely on this flawed concept of regression to the mean, are able to make any money since there is no guarantee of convergence to the house edge (or the probability of a catastrophic event in the case of insurance companies).”
Insurance companies do occasionally fail, AIG comes to mind. Of course, being “too big to fail”, they can rely on a government bailout. Insurance is actually a very complicated business, the wizard of odds is an actuary.
Thanks for the links Kevin and for the insights Kevin and Liz.
If anyone is still interested, do a search for “Eliot Jacobson Getting Lucky at Blackjack”, which would apply to video poker as well, simply different edges and variances.
Also check out “Eliot Jacobson eight common mathematical misunderstandings” where he points out that in a coin flip, heads/tails converges, heads=tails does not, i.e., the ratio 1,000,000/1,010,000 is close to one but 1,000,000 is not close to 1,010,000, it is off by 10,000. 2,000,000/2,015,000 is closer to one, yet it is off by 15,000. When you gamble, the ratio of heads to tails means nothing, but the difference between the number of heads and the number of tails is everything.
“Also check out “Eliot Jacobson eight common mathematical misunderstandings” where he points out that in a coin flip, heads/tails converges, heads=tails does not, i.e., the ratio 1,000,000/1,010,000 is close to one but 1,000,000 is not close to 1,010,000, it is off by 10,000. 2,000,000/2,015,000 is closer to one, yet it is off by 15,000. When you gamble, the ratio of heads to tails means nothing, but the difference between the number of heads and the number of tails is everything.”
Actually the 1st link provided by Kevin Lewis says exactly this (and Kevin alluded to it as well). Except in the link, the statistics compared are “heads minus tails” (basically the same as the statistic heads=tails which you mention) versus heads/tails. The “heads minus tails” statistic is just extremely unstable compared to the proportion statistics – i.e. the variance is enormous relative to the proportion statistic.
So I think this is absolutely true – I totally agree that there is a huge variance difference between these two stats and I appreciate the links and statements. However doesn’t this divergence between the two statistics also apply to the casino ? Why does this divergence only apply to the player ? Maybe if I wear a shirt with the logo “Casino” on it next time I gamble Mother Nature will treat me as she does the casinos.
It applies to the casino as well. Of course the casino has a huge edge with large volume so they are way into many multiples of Nzero with practically zero chance of losing.
“It applies to the casino as well. Of course the casino has a huge edge with large volume so they are way into many multiples of Nzero with practically zero chance of losing.”
As far as I’m concerned, this statement you made above settles the matter, making things perfectly clear –
the difference statistic, A-B (heads minus tails, awaks minus non-awaks, royals minus non-royals, etc.), has an astronomical variance, a player cannot weather the downward, ultra-massive oscillation of this statistic (except maybe a Saudi royalty member), before it begins its massive upward oscillation, the amplitude of the downward oscillations is just too large for a bankroll to remain in tact. It feels great to finally get to the bottom of this, since most of the other explanations never mention the difference statistic, and they just hand wave away the fact that convergence applies not only to the player, but to the casino also – the casino just has the bank roll to weather the variance of the difference statistic and the player doesn’t.
I last received 4 aces dealt with a kicker at Hollywood Casino in St Louis last year on 9/6 JOB. I turned to the guy playing full pay DDB next to me and said, “I believe this was yours.” It wasn’t…I cashed out the $31.25 and decided I needed to do something else for a while.
I have been playing full pay JOB and/or BP in Atlantic City for about 15 years, but probably only 30 hours a year. I feel I play at a relatively high level, but have *never* had a royal flush–either dealt or drawn. With approximately 450 hours of play, does it make sense I’ve never hit the big one? (Unless my math is off, isn’t that around 180,000 hands at, say, at least 400 hands an hour?) Tx for any discussion.
You’re probably dropping the suited ten from hands like AKTs, AQTs, AJTs, KQTs, KJTs, QJTs. It’s easy to do in the glare of the spot lights casinos use to make it difficult to see the display. In any case, you’re not really playing enough to reliably get royals on single line. Even if you play computer perfect, the royal cycle for jacks is something like 40,000 and you need about 5 cycles or 200,000 hands per year or so to avoid the tax penalty. If your goal is to get royals in a short time period, you’re better off with games like 50 or 100 or 250 play. Also consider shorter cycle targets, same as in Keno, if you’re playing 10 spots don’t expect to hit much, conversely if you want to hit stuff, try 4 spots. Keno lets you pick the type of game you want to play, from rare big hits to more frequent smaller hits. Video Keno is often much looser than video slots, progressive and bonusing Keno can present overlays just like lotteries.