Perhaps the Romans misunderstood probability, and thought that adding imperfections to dice would make them more random?

Also the "primes" configuration of faces could be an attempt to minimise bias. A weighted die is less useful if the high numbers are opposite each other.

In the 1560s Cardano's Liber de ludo aleae attempted to calculate probabilities of dice throws. His work was not published until 1663.

There is no earlier known work on the calculus of probability.

Both great points. The thesis of the article (and title!) seems a little premature but interesting stuff.

As long as all players use the same dice, bias may not be that important.

This is a fair point, but I think it gets more complicated if betting becomes involved, because the players may use the same dice, but the player who understands how the dice work has a betting advantage.

Differing beliefs in how games of chance work may still be an arbitrage that works to the advantage of casinos and lotteries.

There was a post on here awhile ago[0] about using loaded dice to cheat at Settler of Catan. All the benefit goes to the person who knows the bias.

[0] https://news.ycombinator.com/item?id=15921322

This makes intuitive sense with just a basic level of understanding.

A biased dice may appear random, but over time youre bound to roll a given number or set of numbers more often. Over time, it adds up to a big advantage.

In many games greater familiarity with the biases of the dice would be no more an advantage (sometimes considerably less) than greater familiarity with the game rules and strategy. It's not clear what games were played, or whether they were played repeatedly amongst groups or by strangers competing against a "house". The article does suggest the biases were visible enough for such games to not hinge upon a particular participant's false believe in the uniformity of the dice though, as well as hint most people's betting strategy probably wasn't sophisticated enough for them to care

This depends on the rules of the game. To use a classic example a coin flip that simply requires HT vs TH (TT or HH get new flips) is unbiased even with a biased coin.

Assuming people had less trust in these dice you end up with different games.

This reminds me of observations some have when looking at the I Ching.

The I Ching is elegant from a modern binary/math perspective because it appears to be an ancient hierarchy of 1-bit symbols (yin/yang), stacked into 3-bit symbols (triagram=element), and finally stacked into 6-bit symbols (hexagram). These are, apparently, descriptions of situations at hand or situations to come. The arrangements of these 64 hexagrams make a modern math mind speculate about binary counting and the reason for a particular ordering [1] and it all seems very orderly and balanced and evenly distributed. Or so you would think.

Then one learns that the traditional methods of division using those 64 hexagrams (e.g. the yarrow stalk method) is quite unevenly distributed. The dice are weighted, so to speak, to give some outcomes more often than others.[2]

Did the Chinese ancients know this and design it this way? Did they care? Was it a mistake? ¯\_(ツ)_/¯

[1] https://www.biroco.com/yijing/sequence.htm

[2] https://aleadeum.com/2013/07/12/the-i-ching-random-numbers-a...

I call BS. The romans (and earlier peoples) were just as cunning and you and me in those matters -- such beliefs didn't play a role in playing other games, and shouldn't play a role in dice.

Those artifacts could be sacrificial or similar special use dice, not something people would use on a game or bet on thinking that some god will even their odds.

That's possible, but it's also possible that they believed in supernatural intervention when playing dice games. It seems they did for other parts of their lives, why would this be different?

Because people in general weren't more stupid than us in everyday matters -- and you don't need a theoretical knowledge of combinatorics to know when you've been had.

Is it really just a matter of "being stupid"?

We know that cleromancy (casting of lots) was practiced among Hebrew populations and in ancient Rome [0]. Under a worldview in which the knowledge and will of the divine is made manifest through natural acts, where things aren't just "happenstance" but are inherently infused with meaning, this makes complete sense.

This is also why trial by ordeal was a thing in many ancient societies, and why it was seen to be legitimate enough to have as part of a judicial system. The idea is that divine forces will show their favor based on the outcome of what would seem to us to be chance. Now there were probably some that were more cynical or less so about all this, and that may be part of how such systems actually gained legitimacy, because those in authority could tip the scales occasionally [1].

I don't think we can compartmentalize the beliefs and mindsets of people, especially ancient people, into "secular" and "religious" domains. It was all intertwined in a way that is hard for us to relate to in modern times, even among modern religious people.

[0] https://en.wikipedia.org/wiki/Cleromancy [1] https://aeon.co/ideas/why-the-trial-by-ordeal-was-actually-a...

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EDIT: I did some more reading and found something interesting -- there is a mention of "loaded dice" in the Mahabharata: [2]

>Shakuni's dice were off course made of his father's thigh bone and is believed to possess that special power of giving the number according to Shakuni's will.

From this the idea that dice could be manipulated or used to cheat in various ways seems to have been known far in the past, but there is still an interesting connection between dice-rolling and supernatural influences, even in the case of loaded dice. One of these days I should learn more about the Mahabharata.

[2] https://www.quora.com/Mahabharata-Hindu-epic-How-did-Shakuni...

Given how bad the modern human is at understanding probability, I see no reason to think the ancients were better.

I agree. And I don't think you need to understand probability to see that a flattened cube will fall on the top and bottom more often than the edges. Another possibility in addition to what you mentioned is that the games the dice were used in took into account that some sides were more probable than others and people trial-and-errored until more-or-less appropriate payoffs/penalties emerged.

These were the points I was about to make.

Also, in modern dice when we want nonequal chance of outcomes we use two dice and sum. This wouldn't be needed if a die had unequal probabilities per face.

"such beliefs didn't play a role in playing other games..."

Source?

I strongly recommend "Against The Gods: The Remarkable Story of Risk" (1996) [1] for a fascinating, historical perspective on people's growing understanding of probability and statistics over the last few millennia.

The author, Peter Bernstein [2], was brilliant and his other works also merit your attention if you are interested in economics and finance.

[1] https://www.amazon.com/Against-Gods-Remarkable-Story-Risk/dp...

[2] https://en.wikipedia.org/wiki/Peter_L._Bernstein

If imperfect dice including one with an extra 3 were found in a big heap, mightn't that be because they were deemed inferior dice unworthy of play? Maybe the local constabulary went around to all the gambling dens, and confiscated & buried all the crooked dice.

> If you thought that your prayers had more to do with a roll’s outcome than anything else, perhaps a perfect shape wasn’t required.

As if someone would play dice (and have fun) with a completely rational mind...

There's nothing preventing someone to play dice and have fun with a "completely rational mind".

Don't confuse accepting small odds and the spending money with being "irrational".

It seems to me that some of the ancients had a better appreciation for chance than we do, especially in public affairs. Many of the speeches in Thucydides' History of the Peloponnesian War caution that though a warring party may be strong, luck is a huge part of war. We hardly have George Bush saying "we are pretty strong, but Iraq could go either way depending on luck." Our politicians speak with such certainty about the future.

> including one cheater’s die with an extra three

I can't imagine that going undetected for long. Which probably means the owner was also capable of sleight of hand.

One would notice on the first roll that the 3 and 4 weren't opposite. Perhaps it's just for a particular game.