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What Makes Dice “Fair?”

What Makes Dice “Fair?”

Look at anything long enough, you’ll start wondering about its particulars, and odds are that more than a few of our readers have stared at a die and genuinely pondered it. Is one face of a 30-sided die more likely to roll up than the other 29? Less likely? Can you actually throw it a certain way to make a favorable roll more likely? Such high-end questions of probability would seem beyond the understanding of the humble gamer.

Well, Numberphile recently turned to Stanford University professor Persi Diaconis to break some figures down into layman’s terms. Some concepts are just a bit too complex to simplify into a bite-sized listicle, of course, so Diaconis’ answers to these questions are appropriately lengthy. At first, he suggests that there are only five polyhedrons (three-dimensional objects) which are actually “fair” in the sense that each of their faces are of equal size, have the same number of vertices, and are just as likely to come up when rolled. So, basically just a cube, a tetrahedron, and a octahedron, then the rarer tongue-twisters that are the dodecahedron and the icosahedron.

Sorry to break the hearts of all the RPGers out there, but by Diaconis’ reckoning, the aforementioned 30-sided die (otherwise known as a rhombic triacontahedron) isn’t actually “fair.” Even if its 30 faces are of the same surface area, they aren’t sharing the same number of vertices (angular points). And there are, in fact, further factors which shape the outcome of a dice role, too. If a six-sided die’s numbers are marked with holes, that can certainly affect their relative weights. The varying amount of paint needed to render all the dots can actually affect the weight, as well. It might just be on an infinitesimal level, but that’s precisely where probability factors in. As Diaconis demonstrates, some die may be unfair just by virtue of allowing “trick throws” that can make one roll more likely.

Being a proper mathematician, though, the professor is willing to reconsider his assertions if challenged. When his interviewer brings up that there may be other criteria by which fairness can be judged and that might expand the selection of fair die beyond the five Diaconis favored, the professor gives the suggestion due consideration. It’s a deep dive into the mathematics of probability, but it’s well worth going through for anybody who wants to answer any arguments at game night.

Still not satisfied? Diaconis goes into even deeper detail with this second video. Among other subjects, he discusses how the wearing-down/rounding-out of dice corners from use can warp probability, how a dice-rolling machine tested that theory, and maybe more interestingly, whether it’s possibly for anybody to “cheat” with dice at a casino (where the house has obviously thought through every angle).

At nearly 21 minutes, this double-sized lecture should hopefully clear up any questions relating to dice. What about cards, though–those other standards of gamefied probability? Presumably, one card in a deck has just as much likelihood of drawing as the rest. Everybody’s heard of card counting at casinos, though. Is one style of shuffling fairer than the next? Turns out this answer is far more “clear cut.”

Could you possibly have any further questions about dice and cards? Have you already learned how to game rolls yourself? Share everything in the talkback!

Featured Image Credit: Numberphile

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