Or, of course…..

But gambling with dice was common in Rome, two millennia ago. There’s something strange about most Roman dice. At first sight they look like cubes, but nine tenths of them have rectangular faces, not square ones. They lack the symmetry of a genuine cube, so some numbers would have turned up more frequently than others.

Even a slight bias of this kind can have a big effect in a long series of bets, which is how dice games are normally played. Only in the middle of the 15th century did it become standard to use symmetric cubes. So why didn’t Roman gamblers object when they were asked to play with biased dice? Jelmer Eerkens, a Dutch archaeologist who has made a study of dice, wondered whether a belief in fate, rather than physics, might be the explanation. If you thought your destiny was in the hands of the gods, then you’d win when they wanted you to win and lose when they didn’t. The shape of the dice would be irrelevant.

The odds adapted to the different probabilities…..

36 thoughts on “Or, of course…..”

  1. As with playing with cards until modern times no-one played with a new set every game.Even absent deliberate marking accumulated wear and tear could be used by those familiar with that set (usually their own) to their advantage.

  2. My oldest brother warned me that the state lottery is a tax on the mathematically challenged.

    I told him buying a lot of tickets was silly. It will take divine intervention to win, so one should be enough.

    He countered that if divine intervention was at play, why buy any? You could find the winning ticket on the sidewalk.

  3. Actually tim the reasoning here is correct. The history of ideas, maths, probability is a fascinating one and would you believe it concepts and behaviours we have today dudnt exist in the past until intellectual achievements in the work of those ideas came about.

  4. Has someone tried out the dice in a systematic manner and determined how significant it actually is?
    A quick google seems to show these dice are still a pretty good approximation of a cube and therefore it can’t be that significant.

    I’m sure if you roll them a million times there is a statistical difference, but given variation in how you hold and throw them and the likely duration of a game (what perhaps 1000 throws?) is it a case that it was still plenty good enough in reality?

  5. “Gamecock

    My oldest brother warned me that the state lottery is a tax on the mathematically challenged.”

    I don’t buy a lottery ticket with any expectation of winning but it does allow me a few moments escape from grim reality to imagine winning.

    £1 – £2 is a small price to pay for that.

  6. Talking of Roman times, there’s a book of jokes from the period. Most haven’t passed the test of time and don’t make any sense but I’ve always liked;

    Barber to customer – “How would you like your hair cut today?”

    Customer to barber “in silence”.

  7. “£1 – £2 is a small price to pay for that.”
    I decided that £1 was a small price to pay for that, but that £2 was too much.

  8. NDReader

    I agree. And since the National Lottery added those extra numbers, I’d won nothing but the occasional ‘lucky dip’ so stopped playing.

    I was going to enter the Postcode Lottery but was advised that winning it was something of a postcode lottery.

  9. The Romans didn’t have a clue about probability and would have put it all down to “the gods”.

    Probability and odds weren’t a thing until some Renaissance Italian genius gambling addict figured it out and used it to beat dice games, iirc.

  10. I just imagine winning – I don’t bother with the faff of buying the ticket.

    Someone did buy me one once. It cost them about £2.50 and I won £5. Not a bad return….

  11. Lots of people these days still don’t get probability. A friend of the Mrs has been developing a dice-based role playing game, and he fundamentally doesn’t understand that two subsequent dice rolls at 60% probability each doesn’t give a 60% probability of success when you have to succeed at both rolls.

    He then blames the players for rolling poorly, and refuses to accept that the probability of success is in fact 36%…

  12. For a couple of weeks at school I became a bookie, taking bets on the school athletics championship. The ease with which I made a useful sum was so absurd that I decided it was indeed like taking sweeties from babies. I never did it again.

    Well, except for a bit of poker in my late teens and early twenties, also a nice little earner. When I was invited to play again in my early thirties I had to be reminded of the rules. Still won though.

    Other than that I once put the rugby team beer money – on which there was a shortfall – into a one-armed bandit. I have never understood the bloody things but won anyway. Our opponents’ jugs were filled. Honour was satisfied. Thanks, Odin.

  13. @DJ

    Think the concept of odds predates probability by a long way since they’re used in many forms of gambling. Of course, until the theory of probability was developed then those odds will not, except by judgment/experience/coincidence, have been mathematically fair… I think Tim’s point stands, that if a dice face was clearly less likely to turn up them the players might have allocated a higher prize to it, for example.

    On the other hand it may have been that people weren’t gambling so much on the dice results themselves, but using them to play a board game (the results of which were gambled on). In that context, small variations in the chances of the numbers you might roll might not have been tactically vital.

  14. The induction of the actual rules as distinct from the ostensible ones is a rich seam to mine.

    With these dice the even probability of each face is the ostensible rule, but how do you get to the real probabilities, and do so in real time in your game?

    Beyond this, in any set of rules, what is to count as cheating, rather than just playing the underlying “real” game? This works both ways of course – Mankad dismissal in cricket.

    For which too see UK’s current politics.

  15. abacab,

    Yes specifically the concept of conditional (or not) probability many people seem to struggle with. Like thinking previous results have an impact on coin tosses or dumb ass rich Chinese studying previous wins at the baccarat table before they bet.

  16. MBE,

    I think I read it in some maths history book, probably that one by Alex Bellis. The story probably is apocryphal I guess.

  17. Dongguan John

    The classic in this field is the Monty Hall problem – to which many Youtube videos give detailed explanations but still confuses many

  18. @DJ

    An early subject of mathematical inquiry into probability was indeed the “fair” division of stakes in unfinished games of chance:


    My point was just that “odds” could be gambled on long before mathematically exact probabilities could be calculated – indeed it’s only fairly recently that odds on sporting events have been subject to extensive mathematical modelling.

  19. @dearieme – You should have stuck with poker as a sideline. Some people make a nice living out of it.

    @DJ – It always makes me smirk to see the Chinese roulette players in Macau with great rolls of paper to record the results.

  20. Bloke in North Dorset

    @ abacab,

    Challenge him to a series of games of backgammon for money. If he doesn’t still get you’ve got a nice little earner.

    @ Ken,

    The Monty Hall problem is covered in one of my bridge (the card game) books. I admit it took a while and a fair bit of scribbling for the penny to drop. It recently came up for discussion on a sailing forum and it was amazing how many people couldn’t see it. Even after I wrote a programme that showed the results some people still wouldn’t accept it.

  21. To explain the Monty Hall problem I tell people to imagine the same exercise with 100 boxes rather than three and it usually clicks then.

  22. The best way of explaining the Monty Hall problem is that if you have initially chosen a wrong door, then changing your choice will result in you winning the big prize. If you have initially chosen the right door, then changing your choice will lose you the big prize.

    And you are twice as likely to have chosen the wrong door.

  23. ‘nine tenths of them have rectangular faces’

    Six faces? If yes, I bet there’s a couple of square faces on each. To wit, I have no doubt a Roman would have looked at it and recognized it wouldn’t land on the end faces as often.

    Just cause we are smarter doesn’t mean they were stupid.

  24. Gamecock, quite.

    If the faces were very obviously not all-square then everyone would know that some faces had less favoured gods, if more subtle then the more perceptive would recognise some numbers were not in the gods best favour on that occasion, and I assume anyone with a die that found favour with a particular number/god would use that whenever they could.

    So were these dice consciously made irregular or did they just turn out that way? Were particular number always allocated to a more or less probable face?

    As to the prospect of winning a lottery, I often view it as I do insurance— a bet that I am better off by losing. A reassuring demonstration that my fate is not the subject of a capricious god but only the chaos of physics. Or, in another mood, that if there is only so much good and bad luck in every fate better not to have squandered it on a mere fortune.

  25. The thing about the lottery is that the odds are so long that your £2 is really only worth a fraction of a penny. But, £2 each week is such a small amount that I wouldn’t miss it. If I extrapolate it to a year or ten years it becomes a significant amount that might bear consideration but against the minute chance that I could become unbelievably wealthy and set up for life.

  26. Malcolm X was a numbers runner in NYC. He said, in Autobiography of Malcolm X, people played the numbers way more than the state lottery because the mob gave much better odds.

    They viewed the STATE as corrupt, not the mob.

  27. The Monty Hall problem is often not described very clearly, which fogs the analysis.

    “You chose a door, and Monty offers another door.”
    Always always always always always always offers another door? Or offers another door when Monty choses to offer another door? Does Monty know what’s behind the doors? Does Monty offer another door when Monty knows you’ve picked a winning door? Or when Monty knows you picked a losing door? I have ***NEVER*** read a description that actually accurately describes the problem.

  28. @jgh: I discovered as a fresher that teachers of probability had a penchant for under specifying the problem. You then had to guess what they had unwittingly left out.

    That discovery lifted a weight from my shoulders – they were being stupid, not me.

    In general, I think, the tricky bit is “modelling” the real world problem. The probability calculations are then usually a piece of cake (at least at the elementary level that mattered to me).

  29. Jgh; Monty always, always, always opens a door. Monty always, always, always, opens the door with the shit prize behind it, and then invites the decision.

    Monty has more information about the game than the player.

    But, yeah. An awful lot of the descriptions given have these ambiguities in them, and people get hung up on trying chase them down.

  30. Bloke in North Dorset


    Malcolm X was a numbers runner in NYC. He said, in Autobiography of Malcolm X, people played the numbers way more than the state lottery because the mob gave much better odds.

    They viewed the STATE as corrupt, not the mob.

    Well the State had to line the pockets of politicians as well as provide them the opportunity to spend money on vote catching white elephant projects. That didn’t leave much for winnings.

    The Mafia were at least honest in only being interested in lining their own pockets, that left more winnings even if the Mafia took a bigger cut than politicians.

  31. The emperor Claudius (who had enough of the god own luck to survive to become emperor), was an inveterate gambler who even wrote a treatise on the subject of “How to win at dice” (which unfortunately has not survived), but it did lead to the scene in the BBC production of “I Clavdivs” where he hands his nephew, the mad emperor Caligula a set of loaded dice which frequently come up “Venus” (a winning throw), whose “luck” he attributed to his own “divinity”.

    I Claudius – 09 – Hail Who @38:35


  32. “I have ***NEVER*** read a description that actually accurately describes the problem.”

    Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let’s say you pick door 1. Before the door is opened, however, someone who knows what’s behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do.

  33. @DJ

    “To explain the Monty Hall problem I tell people to imagine the same exercise with 100 boxes rather than three and it usually clicks then.”

    Ahh and you tell them to imagine Monty opens all the other doors bar one? That’s a great explanation, think even some of the slower wits of our time would realise then.

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