Of Dice & Chance

d20Imagine, if you will, a twenty-sided die (or a d20 as it is known to gamers) being rolled. In the ideal the die has a 1 in 20 chance of rolling a 20 (or any particular number). It is natural to think of the die as being a sort of locus of chance, a random number generator whose roll cannot be predicted. While this is an appealing view of dice, there is a rather interesting question about what such random chance amounts to.

One way to look at the matter, using the example of a d20, is that if the die is rolled 20 times, then one of those rolls will be a 20. Obviously enough, this is not true—as any gamer will tell you, the number of 20s rolled while rolling 20 times varies a great deal. This can, of course, be explained by the fact that d20s are imperfect and hence tend to roll some numbers more than others. There are also the influences of the roller, the surface on which the d20 lands and so on. As such, a d20 will not be a perfect random number generator. But, imagine if there could be a perfect d20 rolled under perfect conditions. What would occur?

One possibility is that each number would come up within the 20 rolls, albeit at random. As such, every 20 rolls would guarantee a 20 (and only one 20), thus accounting for the 1 in 20 chance of rolling a 20. This, however, seems problematic. There is the obvious question of what would ensure that each of the twenty numbers were rolled once (and only once). Then again, that this would occur is only marginally weirder than the idea of chance itself.

It is, of course, well-established that a small number of random events (such as rolling a d20 only twenty times) will deviate from what probability dictates. It is also well-established that as the number of rolls increases, the closer the outcomes will match the expected results (assuming the d20 is not loaded). This general principle is known as the law of large numbers. As such, getting three 20s or no 20s in a series of 20 rolls would not be surprising, but as the number of rolls increases, the closer the results will be to the expected 1 in 20 outcome for each number. As such, the 1 in 20 odds of getting a 20 with a d20 does not mean that 20 rolls will ensure one and only one 20, it means that with enough rolls about 1 in 20 of all the rolls will be 20s. This, does not, of course, really say much about how chance works—beyond noting that chance seems to play out “properly” over large numbers.

One interesting way to look at this is to say that if there were an infinite number of d20 rolls, then 5% of the infinite number of rolls would be 20s. One might, of course, wonder what 5% of infinity would be—would it not be infinite as well? Since infinity is such a mess, a rather more manageable approach would be to use the largest finite number (which presumably has its own problems) and note that 5% of that number of d20 rolls would be 20s.

Another approach would be to say that the 1 in 20 chance means that if all 1 in 20 chance events were formed into sets of 20, sets could be made from all the events that would have one occurrence each of the 1 in 20 events. Using dice as the example, if all the d20 rolls in the universe were known and collected into sets of numbers, they could be dived up into sets of twenty with each number in each set. So, while my 20 rolls would not guarantee a 20, there would be one 20 out of every 20 rolls in the universe. There is still, of course, the question of how this would work. One possibility is that random events are not random and this ensures the proper distribution of events—in this case, dice rolls.

It could also be claimed that chance is a bare fact, that a perfect d20 rolled in perfect conditions would have a 1 in 20 chance of producing a specific number. On this view, the law of large numbers might fail—while unlikely, if chance were a real random thing, it would not be impossible for results to be radically different than predicted. That is, there could be an infinite number of rolls of a perfect d20 with no 20 being rolled. One could even imagine that since a 1 can be rolled on any roll, someone could roll an infinite number of consecutive 1s. Intuitively this seems impossible—it is natural to think that in an infinity every possibility must occur (and perhaps do so perfectly in accord with the probability). But, this would only be a necessity if chance worked a certain way, perhaps that for every 20 rolls in the universe there must be one of each result. Then again, infinity is a magical number, so perhaps this guarantee is part of the magic.

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  1. All fine Mike, by why? This is entry level stats/probability material so it can’t be ‘new’ to you … surely? Yes, the public understanding of stats and probabilities is dismal in the main. This accounts for the widespread incomprehension so apparent in the public’s ‘appraising’ much scientific work, safety advice, risk assessment, climate change modelling, commercial flying etc.

    Quite what chance, ‘luck’ and indeed ‘bad luck’ amount to remains deeply mysterious for many. Humankind, like many other species, seems to have evolved a ‘skill’ in backing ‘best bet’, snap interpretations of how the world around us works. On balance – evolutionary balance – this is clearly a tried-and-tested tactic, especially good for enabling quick decisions that are – most likely – ‘safe’. But the statistics at work here favours the survival of the species, not the individual. However, the startling, often pleasingly confusing, effect of the ‘penny dropping’ in such scenarios can provide us with a powerful lesson too, as well as a laugh. But sad to report, ‘gut feelings’ in these areas are generally strong enough to overcome the cold logic of the statistician and favour the decision for the ‘wrong’ choice of action.

    Your exposition of what might, could, is most likely to, happen with the d20 rolls takes us through how ‘common sense’ interprets such scenarios. But most of us are still rather surprised at that ‘easy money’ bet in a bar with more that 23 people present that – better than 50-50 – two folk will share a birthday date. Poker players know this stuff. Most of us don’t.

  2. I don’t have the link handy, but a physicist built a coin tossing machine, which very precisely tossed the coin, and it came down the same face all the time. It is a physical process, so if replicated closely enough, will give the same result every time. It seems random when we toss the coin because most of us cannot precisely replicate the motion, so there are varying initial conditions. I would expect the same for a 6-faced, or 20-faced, die. Although the more possible outcomes would require more precision to replicate.

  3. An excellent prompt, Keith. You can see the various coin-flipping prototypes at:


    He gets there in the end …. (no 9). A humidity, temperature and air-pressure enclosure would probably get the last few percent of reliability!

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