Blaise Pascal was 17th century genius who invented the mechanical calculator. Pierre de Fermat is famous for a theorem that took three hundred years to prove. What could bring these two minds together? A washed up gambler, down on his luck.
Once upon a time there was a French nobleman named Antoine Gombaud. He lived in the 1600s, before it was possible to accidentally post nude photos on twitter, and so he had to find a different way to destroy himself. After some consideration, he chose gambling. And he chose a particular form of gambling. He calculated the odds of a certain outcome, and bet huge sums based on those odds. This was his game.
Roll a single die four times. What are the odds of getting a six? Well, each roll has a single chance out of six, so four rolls should give us a four out of six chance to roll a six. Good so far.
Now roll two dice twenty-four times. What are the odds of getting a double six? Well, on each roll each die has a one out of six chance to roll a six, but because we have to roll a double six, those two chances have to be multiplied. So each toss of the dice has a one out of thirty-six chance to roll a double six. Roll twenty-four times, and your odds of getting a double six should be twenty-four out of thirty-six, which reduces to four out of six.
Those two outcomes are equally likely, Gombaud thought, and so he set games up based on those odds. After he lost a huge amount of money, it occurred to him that he might have made a little mistake. Did he give up? No. Did he turn to ordinary math students? No. He went to no less a mind than Blaise Pascal.
Pascal was one of those people in history who are annoying accomplished. He was a physician and an inventor who, in his spare time, wrote works of philosophy and clarified physics concepts like the idea of a vacuum. But when it came to Gombaud's little gambling problem, even he was stumped. He kicked the problem over to Pierre de Fermat, presumably sometime before Fermat's Last Theorem.
The two scientists worked the problem over while the gambler fretted. At last they clarified the problem by simply calculating the odds of not rolling a six or a double six.
For the first situation, the total number of possible rolls is calculating by multiplying the six possible outcomes of each roll four times. Six times six times six times six is 1,296. On each roll, there are five possible outcomes that don't include a six, so five times five times five times five, or 625 is your chance of "losing." If you have 1,296 possible outcomes, and 625 times you lose, your total odds of losing are (625/1,296) or around 0.482. So you have a little less than a fifty-fifty chance of losing the game. You're more likely to win than lose.
As we've said before, your chance of rolling a double six during a two-dice throw is one out of thirty-six. That means on every throw there are thirty-six possible outcomes, and thirty-five of them involve you losing. The same rules apply. There are twenty-four games, so you have to multiply thirty-six by itself twenty-four times to get the total number of outcomes. You then have to multiply thirty-five by itself twenty-four times to get the total number of possible losses. Both come up to a substantial amount. In fact, to get the odds of you losing you have to divide 11,419,131,242,070,000,000,000,000,000,000,000,000 by 22,452,257,707,350,000,000,000,000,000,000,000,000. That works out to 0.508, or a little more than a fifty-fifty chance of losing the game. You're more likely to lose than win.
If you are betting using the idea that the one die system has the same odds as the two dice system you are screwed. As Gombaud found out. He learned two valuable lessons: one, it is possible to be a nobleman without thumbs, and two, next time around, ask the famous mathematicians before you set up the games.
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