Short of flipping a coin, it’s hard to think of a more classic representation of “random chance” than rolling dice. But does that hold up? In practice, sure – but technically speaking? Being really nit-picky about things? Nope!
Lucky for you, being unnecessarily nit-picky in a way designed to get you out of family Monopoly night? That’s totally our bag. Here’s why dice rolls aren’t random.
The theory
Bear with us: this is going to get a little philosophical.
Quantum mechanics aside, we basically live in a deterministic world. If we didn’t, whole fields of science wouldn’t be possible: we wouldn’t be able to model the path of a projectile or predict the outcome of a chemical reaction. We’d be flying blind.
And yet, we take it for granted that the outcome of a dice roll is random. Does that make sense?
According to researchers from the Technical University of Lodz, Poland… no, it doesn’t. “The die throw is neither random nor chaotic,” argues their paper, published in 2012. “From the point of view of dynamical system theory, the result of the die throw is predictable.”
Now, sure, practically speaking, that’s not the case – the level of accuracy you’d need to plot out the movements in advance would be virtually impossible to reach. But with enough information, it’s hard to argue that you wouldn’t be able to predict – or even control – the outcome of a dice roll.
“It’s mechanics,” Stanford University math professor (and magician) Persi Diaconis told Numberphile in 2016. “When you release the die from your hand, if you’re actually rolling it, it has velocity, and it has angular velocities, and there’s a phase space. What direction is it going in, and how fast, and then how fast is it turning in each of the various directions?”
There are twelve dimensions of parameters in total that describe the initial conditions of a roll, Diaconis explained – but only six possible outcomes, so that 12-dimensional space gets split into six regions. Whichever region your specific initial conditions fall into, that’s the outcome you’re going to get.
Now, “small changes in the initial conditions, the difference between your hand and your brain, make for a big difference in what side it comes up,” he warned, “because the partitioning of the phase space is finer.” But that doesn’t mean the throw is random – only that it appears that way to us low-information onlookers.
Or, as one 2008 paper put it: “if a dice throw may be taken as a random number generator, this is primarily because of the gambler’s inability to reproduce initial conditions sufficiently well to ensure similar trajectories – and not so much because of an inherently strongly chaotic dynamics.”
The practicalities
So, we’re starting with a setup which, even theoretically, isn’t technically random. But when we take into account certain realities of the world, then things get even worse for the “dice are random” argument.
“When I was a graduate student, we had a guy who was a retired executive who wanted to test the laws of chance,” Diaconis told Numberphile. “He wound up rolling a die three and a half million times, and recording how many times did each face come out.”
And what did he learn from this experience? “The first thing is that dice, if you roll them a lot, get round,” Diaconis explained. “Of course they do. If you roll a die 20,000 times it, you know, bounces around, it changes. So, we had to give him new dice.”
That’s fair, but probably not all that relevant to most people – even at one roll every two seconds, 3.5 million would take more than two years of non-stop throwing to complete. But one factor that is very difficult to swerve is the pips: “a six has six spots in it,” Diaconis pointed out. “Well, those drill holes are lighter. So, the six face actually has less mass, and the one face, which is opposite, has more.”
Even little remnants of the dice-making process can have an effect on how random or fair the resulting dice are. When testing d20 dice, for example, Awesome Dice Blog noticed that the 14 face came up far fewer times than any other.
“We have a theory as to why the 14 rolled so infrequently,” the testers wrote. “Every GameScience die has a small chunk of plastic that sticks out of one face. This flashing is from where the die was removed from the mold […] On GameScience 20-sided dice this flashing is on the 7 face – directly opposite the 14.”
And on that note…
The test
We know what you’re thinking: philosophy and technicalities are all well and good, but surely none of that pans out in real life, right?
It’s difficult to test. You’d need a whole lot of patience, time, and vested interest to justify a proper practical experiment – not to mention a good painkiller for the wrist cramp you’d get from rolling a dice a statistically significant number of times. But… well, that doesn’t mean nobody’s done it.
Enter Awesome Dice Blog. “We used one Chessex d20 and one GameScience d20, both brand new right out of the packaging,” they reported in 2019. “The dice were rolled by hand on a battlemat on a level table. For this experiment the dice were rolled on the surface for at least two feet and had to bounce off a flat backstop before coming to rest.”
“This is similar to the requirements of craps tables in casinos,” the article explains. “Our logic is that if this method successfully prevents cheating with six-sided dice, it will more than suffice for d20 dice being rolled without any intent to alter the results. (Since casinos are not losing money on gambling, we assume they know what they’re doing).”
Twenty thousand dice rolls later, the results were in. So what did they show?
“A casual analysis of the results suggests that neither die is rolling randomly,” the blog concludes.
Indeed, if the dice were truly random, then after 10,000 rolls each – heck, after about 50 – you’d be getting results that were pretty close to an even spread. With a 20-sided dice, that would work out to a nice round 500 rolls of each face – though of course, a dice that came out exactly that even would itself be suspicious.
“We’d expect some deviation,” the blog notes. “Over the course of 10,000 rolls we’d expect, with 85 percent confidence, that each face would be within about 33 of 500 – so anywhere from 467 to 533 is within the bounds of randomness.”
But neither dice got close to this level. In fact, the first dice – the Chessex – only had one value that came up within this range. The second brand rolled more true, with one massive outlier: the 14 came up a whopping 40 percent less often than expected – quite the outlier.
Now, this may have been the deepest practical test we could find, but it wasn’t the widest. That award goes to Mark Fickett’s incredibly extensive and detailed analysis of various dice – which, again, found very few whose rolls seemed truly random.
So, what to do if you want a truly random dice game? Your options are limited, we’re afraid: you can either go to Vegas, where they work hard to make sure they have the fairest dice possible… or you can just make like the Romans, and leave it up to the gods.
