## Sunday, June 9, 2024

### "Rolling the Dice: What Gambling Can Teach Us About Probability"

From LitHub, May 23:

Tom Chivers on the Historical Origins of the Science of Statistics

Traditionally, the story of the study of probability begins in French gambling houses in the mid-seventeenth century. But we can start it earlier than that.

The Italian polymath Gerolamo Cardano had attempted to quantify the maths of dice gambling in the sixteenth century. What, for instance, would the odds be of rolling a six on four rolls of a die, or a double six on twenty-four rolls of a pair of dice?

His working went like this. The probability of rolling a six is one in six, or 1/6, or about 17 percent. Normally, in probability, we don’t give a figure as a percentage, but as a number between zero and one, which we call p. So the probability of rolling a six is p = 0.17. (Actually, 0.1666666… but I’m rounding it off.)

Cardano, reasonably enough, assumed that if you roll the die four times, your probability is four times as high: 4/6, or about 0.67. But if you stop and think about it for a moment, that can’t be right, because it would imply that if you rolled the die six times, your chance of getting a six would be one-sixth times six, or one: that is, certainty. But obviously it’s possible to roll six times and have none of the dice come up six.

What threw Cardano is that the average number of sixes you’ll see on four dice is 0.67. But sometimes you’ll see three, sometimes you’ll see none. The odds of seeing a six (or, separately, at least one six) are different.

The idea is not to look at the chances that something would happen by the number of goes you take, but to look at the chances it wouldn’t happen.

In the case of the one die rolled four times, you’d get it badly wrong—the real answer is about 0.52, not 0.67—but you’d still be right to bet, at even odds, on a six coming up. If you used Cardano’s reasoning for the second question, though, about how often you’d see a double six on twenty-four rolls, it would lead you seriously astray in a gambling house. His math would suggest that, since a double six comes up one time in thirty-six (p ≈ 0.03), then rolling the dice twenty-four times would give you twenty-four times that probability, twenty-four in thirty-six or two-thirds (p ≈ 0.67, again).

This time, though, his reasonable but misguided thinking would put you on the wrong side of the bet. The probability of seeing a double six in twenty-four rolls is 0.49, slightly less than half. You’d lose money betting on it. What’s gone wrong?

A century or so later, in 1654, Antoine Gombaud, a gambler and amateur philosopher who called himself the Chevalier de Méré, was interested in the same questions, for obvious professional reasons. He had noticed exactly what we’ve just said: that betting that you’ll see at least one six in four rolls of a die will make you money, whereas betting that you’ll see at least one double six in twenty-four rolls of two dice will not. Gombaud, through simple empirical observation, had got to a much more realistic position than Cardano. But he was confused. Why were the two outcomes different? After all, six is to four as thirty-six is to twenty-four. He recruited a friend, the mathematician Pierre de Carcavi, but together they were unable to work it out. So they asked a mutual friend, the great mathematician Blaise Pascal.

The solution to this problem isn’t actually that complicated. Cardano had got it exactly backward: the idea is not to look at the chances that something would happen by the number of goes you take, but to look at the chances it wouldn’t happen.

In the case of the four rolls of a single die, your chance of not seeing a six on any one throw is 5/6, or p ≈ 0.83. If you roll it again, your chance of not seeing a six on either throw is 0.83 times 0.83, or just shy of 0.7. Each time you roll the die, you reduce the chance of not seeing a six by 17 percent.

If you roll the die four times, your chance of not seeing a six is 0.83 × 0.83 × 0.83 × 0.83 ≈ 0.48. (To save time, we can say “0.83 to the power 4,” or “0.83 ^ 4.”) So your chance of seeing a six is 1 minus 0.48, or 0.52, or 52 percent. If you bet at even odds one hundred times, you’d expect to win fifty-two times, and you’d be in profit.

But look what happens when we do it with the two dice, looking for a double six. Your chance of seeing a double six on one roll of two dice is 1/36, or p ≈ 0.03, as we said earlier. So your chance of not seeing a double six is 35/36 or about 0.97.

If you roll your dice twenty-four times, your chance of not seeing a double six is 0.97 multiplied by itself twenty-four times (0.97 ^ 24). If you do that sum, you end up with 0.51. So the chance of seeing a double six is 0.49. If you bet at even odds, you’d expect to see it forty-nine times in a hundred, and you’d lose money.

(We should take a moment, here, to recognize the absolutely heroic amount of gambling that Gombaud must have been doing in order to be able to tell that his 52 percent bet was coming off, but his 49 percent bet wasn’t. Apparently, he had deduced, correctly, that you need twenty-five rolls of the dice, not twenty-four, for it to be a good bet. Gombaud was a man who enjoyed his dice-rolling.)

This led Gombaud to raise another question with Pascal. Imagine two people are playing a game of chance—cards or dice. Their game is interrupted halfway through, with one player in the lead. What’s the fairest way to divide the pot? It seems wrong to simply split it down the middle, since one person is winning; but it’s also unfair to give it all to the player in the lead, since they haven’t actually won yet....

....MUCH MORE

All of which leads, 300 years later, to:

Saturday, 11 July, 1959: 2:07 A.M.
I am awake and alone at 2 A.M.
There must be a God. There cannot be a God.
I will start a blog....