One of my stat books says.
Event F is getting double sixes and event E is not.
Nitpick: There were actually two problems that Chevalier de Méré asked about. One was about the probability of rolling double sixes in 24 rolls. The other was the “problem of points,” which was about the fair distribution of stakes when a game is interrupted before it’s finished.
There is no doubt in my mind that basic probability has been understood since gambling was invented.
Honestly, it doesn’t take much brains to know there’s a 50/50 chance on a coin toss, even if you didn’t express it as such.
But bear in mind that insurance also has a negative expected value to the purchaser. Utility and simple probabilistic expected value are not the same thing.
Yes, in fact, with fair dice, 24 is the largest number of rolls that still gives the house an advantage. I suppose this game developed via trial and error, but it’s interesting.
Yes, with 25 rolls, the player has about 50.5% chance of winning. Seems like a rather tedious game if one play consists of 24 rolls of the dice.
I would hardly call it an unfair bet. The house has a slight edge, but that’s true of all gambling. It’s about the same as betting on red/ black at roulette with a single zero on the wheel.
I agree it sounds like a tedious game. Was there more to it?
It’s already been indirectly cited, but Hacking’s The Emergence of Probability really has been the crucial treatment of the subject that all the academic discussion about the issue since has either been the reactions for or against. The utterly essential interpretation.
Broadly, I tend to agree with him. It was ultimately a weirdly (uniquely?) original shift in how humans could think about the world.
Perhaps the rising tension as one got closer and closer to the 24th roll was enough to make it exciting.
Or maybe they were just easily amused in Renaissance France
The ancient Chinese gambling game Pai Gow — related to Baccarat but much more complicated — was somehow analyzed over the centuries to come up with “best” play (though a single best strategy was never agreed). This is quite difficult; for one thing, like Rock-Paper-Scissors, optimal play is strongly dependent on opponent’s strategy. When the game was introduced to Las Vegas (in the 1970’s?) different casinos had slight differences in the ‘House Way’ — the mandatory strategy when casino money was backing a hand.
Caesar’s Palace, however, had a bizarre stupidity! For example, holding a Six, a Four and the two Je Joon tiles it is overwhelmingly best to set them as (9,7) — quite likely to win and extremely unlikely to lose. Caesar’s Palace House Way however was to set them as (Je Joon, 0) — almost certain to push (break even). Their rule was to never split the Je Joon pair since it is impossible to lose with Je Joon. :smack:
There were probably side bets at increasingly mounting odds. By the time the 24th roll approached the banker betting against 6&6 might be forced to pay out hugely if shooter suddenly got lucky. He might shout “insurance”, e.g. offering to take 25-to-1 odds on 6&6 as a hedge to protect the large pile of cash collecting in front of him.
As mentioned upthread, these side-bets, and requests for insurance (hedging) bets, and equitable pot splitting probably motivated some of Cardano’s own study.
I’ve witnessed this scenario at least once in a high-stakes backgammon game. On the very last shake opponent wins only if he throws 6&6 — but if he does you’re busted, perhaps going deeply in debt. Wouldn’t you halt the game and shout “Anyone offering insurance?” ?
It’s also incredibly sensitive to the accuracy of the dice. The game would be fair if double-sixes came up 1.025 times out of 36. So even a tiny push could send it in favor of the player. I’m not sure that even modern casino dice are that accurate.