It’s a rainy Wednesday night in 50 BC, and you’re playing Yahtzee against Julius Caesar. You have four 4s, one 6, and one roll left. You have a decision to make: Do you risk it and try to get a Yahtzee, or settle for a fairly impressive four-of-a-kind? It turns out this question—of calculating the odds of whether or not a die will do you a favour—is not only an incredibly ancient question, but the basis of probability calculus. 

In a recent paper published in International Statistical Review, McGill professor Christian Genest explores the historical role of dice in the development of probability calculus. By exploring the shifting role of dice throughout history, Genest and his co-author David Bellhouse bring preexisting theories together, creating a comprehensive perspective on the evolution of probability calculus in the Western world. 

Dice have long been regarded as central to the development of probability calculus; they allow for randomness to be physicalized in a simple, predictable, and empirical way. However, while there is evidence of ancient Mesopotamians using dice in their games, the first probabilistic calculations only date back to the 13th century. 

“The comparatively late emergence of this mathematical concept is somewhat surprising, given that humans have been confronted to randomness since time immemorial,” Genest explained in a written statement to The Tribune. 

Genest and Bellhouse attribute this late development to a number of conditions present in the Roman empire, one such factor being the lack of unbiased dice. 

Astragali or tali, made of the ankle bones of goats or sheep, were common substitutes for dice. They had four playable sides and worked as modern dice do, but lacked any regularity that would allow for statistical data to be consistent from one talus to another.

“To my surprise, however, this fact did not appear to be so well been documented in the literature, so with my son Richard (who was 9 at the time), I endeavored to toss several modern tali (which we got from a butcher) hundreds of times each in the summer of 2024,” Genest wrote. “We had a lot of fun doing this together.”

The variation in the data they collected led Genest and Bellhouse to conclude that any probabilistic calculations would have been difficult to make and impossible to generalize. However, Genest suspects the Romans never even got that far. 

“People [had a] tendency in ancient times to hold a deterministic view of the world that led them to interpret the result of random events, such as the throw of dice, as a manifestation of some deity’s will,” Genest explained. 

Combined with their deterministic perspective, Genest explained that the use of Roman numerals would have hindered the Romans’ ability to complete any significant calculations; you can’t do any serious math with Roman numerals.

It follows that the Western discovery of probability calculus followed the implementation of Arabic numerals in Western society. In fact, the earliest found source of combinatorial calculations—a Latin poem “De Vetula,” published in the 13th century—is believed to be, in part, a way of introducing readers to Arabic numerals.

Together, the implementation of Arabic numerals, the creation of more “regular” dice, and the fallaway of determinist perspectives allowed for probability calculus and combinatorial mathematics to develop in the West.

“While ‘De Vetula’ seems to be the oldest Western source to date, it is entirely possible that traces of probability calculations could be found in older literature from Chinese, Indian, or Arabic culture. We need to look into it!” Genest wrote. 

This goes to show that while we can hedge our bets on the odds of rolling that last 4 we need, if the Romans’ had Yahtzee, they wouldn’t have even known they had bets to hedge; their dice were biased, and this bias, instead of being attributed to physical structure, was attributed to the will of the gods.

So, yes, if you played Yahtzee against Julius Caesar using modern dice, it would be reasonable to believe—statistically speaking—that you would win, regardless of whether or not you try for that five-of-a-kind.