William M. Briggs

Statistician to the Stars!

What Is A Game Of Chance?


This originally ran 4 December 2015, but since we need it for tomorrow’s crucial article on whether fantasy sports are “games of chance”, it is best to refresh our memories. This is also complementary to yesterday’s post.

What is a game of chance? There is no such thing as chance. Chance does not exist, though many think it does. Given that the term crops up repeatedly, and people do take meaning from it, we have to figure what it is that people think the term means with respect to gambling. Best guess, as I’ll show, is that chance appears to be as a synonym of mostly not predictable. “Games of chance”, therefore, could be translated as “games which are mostly not predictable”.

Take craps. On the come-out, the shooter wins with a 7 or 11. The two-dice total will come to something, possibly this 7 or 11. Is this total “chance”? Well, craps is taken by the law (and by everybody else) as a game of “chance.” But what is really meant is that that outcome of the roll is not predictable beyond the constraints that the two-dice total must fall in the range 2 to 12. This knowledge of the bounds is a form of predictability, albeit a weak one.

If the only—pay attention here: I mean this word in its literal sense—information that we have is that “There will be a game played which will display a number between 2 and 12 inclusive”, then we can quantify our uncertainty in this number, which is that each number has probability 1/11 of showing (there are 11 numbers in 2 to 12).

Craps players have more information than this. They know the total can be from 2 to 12, but they also know the various ways how the total can be constructed, e.g. 1 + 1, 1 + 2, 2+ 1, …, 6 + 6. There are 36 different ways to get a total using this new information, and since some of the totals are identical, the probability is different for different totals. For example, snake eyes, 1 + 1, has 1/36 probability; there are 6 ways to get a total of 7, for a probability of 1/6.

It should be clear that these different probabilities are not a property of the dice (or the dice and table and shooter, etc.). If probability were a physical property, then it must be that the total of 2 has 1/11 physical probability and 1/36 physical probability! How does it choose between them? Quantum mechanics? (There is a big hint here about interpreting QM, which I’ll skip today.) No, probability is a state of mind; rather, it states our uncertainty given specific information. Change the information, change the probability.

There are any number of physical mechanisms that cause each dice total, causes of which we are mostly or completely ignorant. We know the causes must be there, we just don’t know what they are. We do know there are many causes: imagine the bouncing rolling dice flopping around, buffeted by this and that. If we knew some of these causes for individual rolls—perhaps we could measure them in some way as the dice fly; say, by noting the walls of the table are cushier and more absorbent than usual—then we could incorporate that causal information and, again, update the probabilities of the totals. A 7 might be more or less probable depending on hos the information “plays”.

Casinos base their craps payouts on their “vig” (or cut, or “percentage”) and on the probabilities calculated only using the information of how the totals are reached. If you had extra information about causes, you could use that to “beat” the system—assuming their vigorish is not too vigorous; it’s the transaction fees that always kill you. Unless your knowledge of cause is complete, you might not necessarily beat the casino for any single game, but if you have good causal knowledge, you will beat them over multiple games. It is for this reason that casinos ban contrivances that could measure causes or proxies of causes.

Nobody could, in the scenario of a casino (but in a physics lab, the situation would be different) measure all causes of a dice roll; but to win consistently, all causes don’t need to be measured, just some of them, or their proxies (things related to a cause which is measurable). It is the measurement and knowledge of cause, and not just bounds, that requires skill and turns “games of chance” into “games of skill.”

Think of it this way: take two craps players, one a novice but who knows the rules, and the other an expert who claims, falsely, that he is able to measure some of the relevant causes. Pitted against one another, each is as likely to win (more money) as the other. But if the second player truly can measure some of the relevant causes—perhaps he is a physicist with secret measurements devices which allow him to know some but not all causes—he will beat the first fellow consistently. How consistently depends on the extent of his causal knowledge.

We have arrived at a better definition of “game of chance”. It is “game where the causes are unknown but where the outcomes are defined”. Once any of the causes become known, the game becomes, at least partially, a game of skill.


  1. Even roulette could be eventually predicted if one could measure the physics of the wheel, the throw, etc. I wonder about things like “Google glasses” (especially when the pair looks identical to “regular” glasses) and how that will affect gambling. Sooner or later, our clever tech people will make games of “chance” more or less obsolete as they devise ways to measure the causes. Not tomorrow, of course, but eventually.

  2. If the only—pay attention here: I mean this word in its literal sense—information that we have is that “There will be a game played which will display a number between 2 and 12 inclusive”

    So the card counter can reduce the number of values that he’s working with, there is still “chance” or uncertainty, the actual card he receives is still NOT predictable, he can simply play with different odds. (The fact that he can win more than lose has nothing to do with any one outcome of the game.)

    Couldn’t some casino (instead of kicking out card counters), create a “game” for them in which the house would still mostly win, except for the very best counters?

    That sounds like a “draw” to me for all the geniuses out there to put up or shut up.

  3. Briggs

    December 3, 2015 at 11:32 am

    John B(),

    I would use different language and say instead that the card counter increases his predictive ability. His odds are different than the non-card-counter because his information is different. There are no intrinsic odds; all odds are a function of exactly specified information.

    Casinos take action against card counters by using multiple decks, which reduces the juiciness of information garnered from counting. But they still have small-deck Blackjack, for instance, though the raise the minimum bid to compensate.

  4. Once any of the causes become known, the game becomes, at least partially, a game of skill.
    Something about this sentence doesn’t sit well. Let’s say one of the faces on one of the dice has a probability of > 1/6 (meaning the other faces have probabilities of < 1/6). One player knows this and the other doesn't and so the first becomes more "skillful" when he applies this knowledge to betting. When the second player learns the information does the first become less skillful? Does skill here mean unequal application of special knowledge?

  5. Briggs

    December 3, 2015 at 11:53 am


    Ah, but how is it that one of the faces “has” probability > 1/6? There is no such thing as probability, so in order for this to be true, we must be imagining some evidence about how that particular die will roll, which is causal knowledge.

    Make sense?

    Addendum: by “skill” I mean the very specific sense of knowing or measuring cause. Maybe not the best word for it, but I’m trying to fit this into the legal scheme of things (for outside reasons).

  6. Ye Olde Statistician

    December 3, 2015 at 1:52 pm

    Many years ago, in the Great State of Colorado, the Democratic Party became a Church. By this, I mean the Party began running weekly Bingo games as fundraisers. The Jefferson County Party raked in the bucks because while folks might not like the Party’s platform, they liked playing Bingo. The Other Party soon started their own games as did any number of charitable and fraternal organizations.

    Then the Attorney General of the State declared at a Press Conference that Bingo was a Game of Chance, and these were illegal under current law. (This was before the State got into the Numbers Racket by running lotteries.) In the chit-chat following the Press Conference, one of the reporters commented to the A/G that poker was not entirely chance, since there was some skill required in “reading” other players, deciding which cards to trade in, how much to wager, and so on. The A/G agreed that poker was indeed a game of skill.

    You know what happened next. Poker parlors bloomed like the Mountain Marigold from the Prairie to the High Country and beyond. Heh.

  7. I keep wanting to +1 posts here…

    +1 to YOS.

    All markets are free markets.

  8. EDITED:

    “…a better definition of “game of chance”.

    It is “game where the causes [of what?– ‘of any specific outcome’ ?] are unknown but where the [range of possible] outcomes are defined”. Once any of the causes become known, the game becomes, at least partially, a game of skill.

    Something’s missing — consider Keno, where one can track which specific balls get sucked into tubes to achieve each score. Every so often they wear unevenly, or whatever, and certain balls start getting picked up more often & other never or next to never. In such a situation, over an hour or two one can get a very high probability of selecting all numbers that are not picked (at one time/casino a $1 or $2 bet in Vegas could result in a $1000 payout by picking a series of numbers none of which are picked; but only $12 for only one selection & no more being picked).

    The point being, one might have no clue what a particular “cause” is, but have sufficient info (history of recent picks) to know that some significant randomness is removed — there’s a “cause” lurking there that can be exploited. The definition involving “cause” seems to need some work.

  9. Briggs

    December 3, 2015 at 2:07 pm


    Yes, it’s not cause per se, but knowledge of cause, or causal knowledge, which is why the language of proxies and so forth. That kind of knowledge is present if you have measured, or have other reason to believe (which is information), of changes in the causal path. That’s true for worn balls.

  10. YOS,

    Perhaps not coincidentally, that’s the same reason that various weekly fantasy football sites use to not be subject to gambling law. Since players get to choose their actions it is called a game of skill.

    Unfortunately for them, but good for the lawyers, there is still a large amount of uncertainty, which can be used to argue that it’s a game of chance. Perhaps our legal better can settle the difference of chance and skill for us. Time will tell!

  11. James,

    New York’s Attorney General just declared fantasy sports sites illegal last week. He has ordered a cease and desist against two companies. It would appear his concern is with the daily fantasy sports betting. It seems the state’s view is this activity is not skillful whereas the longer term monthly and seasonal bets are skillful. It will be interesting to see how this plays out.

    Some of the sports leagues (MLB included) are in partnership with some of the Fantasy Sites and have declared that they are games of skill and are therefore not gambling. I suspect this is a political move or one designed to bring the state some sort of payoff for allowing the activity. After all, the state is itself in the gambling business with its lotteries.

  12. Sherri,

    In the 1960s, Claude Shannon and Edward Thorpe build a computer to beat roulette. Shannon was already the father of information theory. Thorpe would literally write the book on card counting, and then go on to be a very successful hedge fund manager.

    Every croupier has a “style”, a degree of firmness with with he flicks the ball into the track. And it fairly regularly drops out of the track at about the same time with each flick. And the rotation of the wheel is pretty stable, too.
    Most of the time, the ball hits one of the deflectors before dropping into the wheel, but often enough it doesn’t. With knowledge of the time it takes for the ball to drop, the rate of rotation of the wheel, and the position of the wheel when the croupier flicks the ball, it is easy to find a section of the wheel more likely to receive the ball than others.

    In Las Vegas, it is illegal to use technology to aid your gambling. But counting is legal. However, the casino has the right to refuse service to anyone. So, if you are too good you will be barred.

    And Thorpe went on to hedge funds because there is more money to be made on Wall St. than in Vegas.

  13. What is a game of chance?

    The game one happens to be playing. 😉


  14. I’ll start off with a quote that expresses my own opinion about defining “chance”:
    “When I use a word,” Humpty Dumpty said, in rather a scornful tone, “it means just what I choose it to mean—neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master—that’s all.”
    Lewis Carroll, Through the Looking Glass.
    I think Brigg’s notion that chance refers to unpredictability is a good base for the definition. It agrees with that in the Stanford Encyclopedia of Philosophy article on Chance and Randomness (see:
    and Futuyma’s definition:
    “scientists use chance, or randomness, to mean that when physical causes can result in any of several outcomes, we cannot predict what the outcome will be in any particular case.”
    The article goes on to distinguish between chance and randomness.

    One point which has been touched on in several comments is whether, given sufficient knowledge of all the initial physical conditions, what one might regard as unpredictable (e.g. the toss of a die), might be predictable. I’m not sure that would always be possible in classical physics, given situations where chaotic conditions might prevail, and it certainly wouldn’t be possible in quantum mechanics.
    By the way, I wonder if anyone has thought of making a game of chance out of the double-slit experiment (or a simulation thereof)? I doubt that would be subject to house manipulation.

  15. Briggs

    December 3, 2015 at 4:29 pm

    Steve E,

    This was my motivation for starting down this road. More on this at a later date. I have an interest in this.

  16. Steve E,

    I had been watching the case with some interest, but spaced out and didn’t realize that the ruling had come down!

    I agree that it is mostly about the state getting its own vig from the activities. The government has never been against gambling, it’s just been against not making as much money as the casinos. Besides state attorney generals, I have been hearing that various casino owners want this regulation as well. It might even become another bootleggers and baptists, which should be fun for everyone.

    I wonder what Brigg’s interest in this is? Is he secretly raking in cash from online fantasy games because big oil hasn’t yet sent him a check? Perhaps someone has asked him to quantify skill vs chance?

  17. James:

    Sadly, if our host had a piece of the rake I’m sure his blogging days would have ended. I’m sure (by which I mean I hope) it’s something much more diabolical. Your latter suggestion may be more likely.


    I listened to a very interesting discussion on this issue on sports talk radio last week and much was being made about “skill” and that it seemed absurd to suggest that it could be applied in the long term but not the short term. Not all of the participants were in agreement of course. Still, it was one of the more intelligent discussions I’ve heard on sports talk radio.

    I look forward to your future post.

  18. May the force be with you!

  19. John B writes “If the only information that we have is that There will be a game played which will display a number between 2 and 12 inclusive”

    Then the game could deliver a “2” on every single round. No assertion to gaussian or uniform distribution.

  20. JohnB,

    “Couldn’t some casino (instead of kicking out card counters), create a “game” for them in which the house would still mostly win, except for the very best counters?”

    1, Card counting only works for one specific game, Blackjack.

    2. The casinos already have a method to beat card counters. It’s called a five deck shoe. Casino black jack dealers no longer deal from a loose deck of cards, they haven’t worked that way for decades.

    The shoe is a box containing shuffled cards set up so that the dealer can only draw the top card. Shoes were originally created for a single deck to prevent the dealer from assisting in a cheat by dealing off the bottom.

    When the shoe ran out the dealer would re shuffle the deck and put it back in the shoe.

    When the Casinos first caught on to card counting, they stared throwing them out of the casino But eventually the figured out that they could beat the card counters by using a multi deck shoe. They currently favor 5 deck shoes. for the 5 deck shoe, five decks are randomized and shuffled together by machine. It has supposedly been proven that no card counter can beat a 5 deck shoe without the assistance of a computer.

  21. I recall that Darrell Huff’s “How to take a chance” had a cartoon with this text:

    Cousin Zeb: Uh, is this a game of chance?
    Cuthbert J. Twillie: Not the way *I* play it, no.

  22. No such thing as chance? Pasteur said it favored the prepared mind.

  23. Michael 2,

    Indeed it could deliver a 2 every time, but we aren’t assuming any distribution. There are 11 possible results (that is all we know). Briggs wrote a whole paper on how this is logically sound.


  24. There’s only 2 ways to actually win in a casino. First, own the casino. Second, get free chips. It does happen. See Paschal’s Wager for more on this offer. It’s in his book, ‘Pensees’ to see if you qualify for them. Or read ‘The Barbarian Bible’ on how to use them.

  25. Pasteur wasn’t laying odds in Vegas.

  26. Sorry to be a bit dense: I don’t understand that that given:
    “There will be a game played which will display a number between 2 and 12 inclusive”
    It follows that:
    “we can quantify our uncertainty in this number, which is that each number has probability 1/11 of showing”
    It is true that “there are 11 numbers in 2 to 12” but the game could involve the throw of a dodecahedron with faces marked 2..12 and 12 on the remaining face. In this case each number would not have a probability of 1/11 of showing. Or the game could involve some other scheme with non-equal representation of the numbers that can be displayed.
    Shirley, all one can say is that the probability of display of each number is >0 and <1 and can be expressed as a vulgar fraction.
    Please explain…

  27. …or maybe not even a vulgar fraction: the “rules” include the case where the output is some non-linear function expressed as mod(11)+1. In that case all one can say is 0<p<1 ???
    Why am I wrong?

  28. May be slightly OT (but maybe not, since it’s about statistics), but here’s something I’ve never heard of before:


  29. @Phil R

    Woop Woop SJW Warning!

    “Using statistics to support progressive social change”

    Oh dear – sounds like “how to fiddle the figures & cook the books: A guide for SJWs”

  30. Gareth,

    That’s what I was thinking. I followed a link in a Delingpole article that references a Dr. Rachel Cohen. her description is:

    Dr Cohen’s main interests are in the sociology of work and employment; especially ‘non-standard’ work, including self-employment, mobile work, and homeworking and in work-life boundaries. Her PhD focused on the working lives and employment relations of hairdressers. Her current research explores similar issues in the working lives of car mechanics and accountants. Her research involves a mixed-methods approach.

    “Dr Cohen has co-edited a special issue of Sociology of Health and Illness on ‘body work’ (work which takes the bodies of others as its object), and an issue of The International Journal of Social Research Methodology on feminism and quantitative methods. Both of these form part of ongoing collaborative projects. Dr Cohen is interested in statistical literacy and is currently editor of Radical Statistics (www.radstats.org.uk). She has also written on gender and sport.”

  31. gareth,

    It’s been years since my college days, and I only studied pure math, not statistics, so I could be misunderstanding Briggs’ argument, but let me give it a go.

    When you say “It is true that “there are 11 numbers in 2 to 12” but the game could involve the throw of a dodecahedron with faces marked 2..12 and 12 on the remaining face” you are making Briggs’ point but with a different set of causal inputs. He started with the initial piece of information “(integral) numbers from 2 through 12” but that was all. He later added the bit about the two (presumably fair) dice, but that’s at the point in his argument that you inserted your dodecahedron instead. So just follow his argument, but with your different set of known probabilities. And the knowledge that you’re not in Vegas, because they don’t use dodecahedra.

  32. gareth,

    What you are doing wrong is adding information. You can do this all day, for example, suppose the game were to be played under water and the game objects were constructed of sugar or suppose the game is rigged or … ad infinitum. Each would have its own probability distribution. However, if all you know is that there are 11 possible outcomes your uncertainty in which will be next must be equally divided among all of them and then each must be 1/11. When you get more information then your uncertainty in the outcomes will likely change.

  33. gareth,

    See Briggs’ paper for more details. The argument is on pages 25 – 27.

  34. gareth – Sorry, I was just flipping through again and he actually makes the argument much earlier – those examples are mathematical proof attempts.

  35. How does “similarity” factor into probability?

    Meaning, no two coin flips are exactly identical. If we flip a coin 100 times it will land heads” 50% of the time, but there is a 100% chance that each flip will be a unique, existential event. The similarity between flips is epistemic, not existential.

    What is the probability that two coin flips will rotate the same number of times on the way up, the same number of times on the way down, will bounce off the ground the same number of times and roll in a counter-clockwise direction and hit the floor board and land heads?

  36. DAV, nate, I’m 1000 miles from home, reading on a 7″ tablet which won’t show pdf so can’t read the paper. I don’t think I was *adding* information – quite the opposite. I was pointing out that we don’t know the probability is equal for each value. It might be unequal, we do not know. Either it is or it isn’t – and there is no information in “A or Not A” statements.

Leave a Reply

Your email address will not be published.


© 2016 William M. Briggs

Theme by Anders NorenUp ↑