The Dice Problem

Thinking in probability can be hard sometimes. Thinking of probability in a wrong way can be costly, especially at the casino. This was what happened with Chevalier de Méré (1607-1684), who was a French writer and apparently an avid gambler. He estimated that the odds for winning in this one game were in his favor. However, he was losing money consistently on this particular game. He sensed something was amiss but could not see why his reasoning was wrong. Luckily for him, he was able to enlist two leading mathematicians at the time, Blaise Pascal and Pierre de Fermat, for help. The correspondence between Pascal and Fermat laid the foundation for the modern theory of probability.

Chevalier de Méré actually asked Pascal and Fermat for help on two problems – the problem of points and the dice problem. I wrote about these two problems in two blog posts – the problem of points and the dice problem. In this post, I make further comments on the dice problem. The point is that flawed reasoning in probability can be risky and costly, first and foremost for gamblers and to a great extent for anyone making financial decisions with uncertain future outcomes.

For Chevalier de Méré, there are actually two dice problems.

The first game involves four rolls of a fair die. In this game, de Méré made bet with even odds on rolling at least one six when a fair die is rolled four times. His reasoning was that since getting a six in one roll of a die is $\frac{1}{6}$ (correct), the chance of getting a six in four rolls of a die would be $4 \times \frac{1}{6}=\frac{2}{3}$ (incorrect). With the favorable odds of 67% of winning, he reasoned that betting with even odds would be a profitable proposition. Though his calculation was incorrect, he made considerable amount of money over many years playing this game.

The second game involves twenty four rolls of a pair of fair dice. The success in the first game emboldened de Méré to make even bet on rolling one or more double sixes in twenty four rolls of a pair of dice. His reasoning was that the chance for getting a double six in one roll of a pair of dice is $\frac{1}{36}$ (correct). Then the chance of getting a double six in twenty four rolls of a pair of dice would be $24 \times \frac{1}{36}=\frac{2}{3}$ (incorrect). He again reasoned that betting with even odds would be profitable too.

The problem was for Pascal and Fermat to explain why de Méré was able to make money on the first and not on the second game.

The correctly probability calculation would show that the probability of the event “rolling at least one six” happening in the first game is about 0.518 (see here). Thus de Méré would on average win 52% of the time in playing the first game at even odds. In playing 100 games, he would win about 52 games. In playing 1,000 games, he would win about 518 games. The following table calculate the amount of winning per 1,000 games for de Méré.

Results of playing the first game 1,000 times with one French franc per bet

 Outcome # of Games Win/Lose Amount Win 518 518 francs Lose 482 -482 francs Total 1,000 36 francs

Per 1,000 games, de Méré won on average 36 francs. So he had the house edge of 3.6% (= 36/1000).

The correct calculation would show that the probability of the event “at least one double 6” happening in the second game is about 0.491 (see here). Thus de Méré could only win about 49% of the time. Per 1,000 games, de Méré would win on average 491 games, or the opposing side would win about 509 games. The following table calculate the amount of winning per 1,000 games for de Méré.

Results of playing the second game 1,000 times with one French franc per bet

 Outcome # of Games Win/Lose Amount Win 491 491 francs Lose 509 -509 francs Total 1,000 -18 francs

The winning on average for de Méré is negative 18 francs per 1,000 games. So the opposing side has a house edge of 1.8% (= 18/1000).

So de Méré was totally off base with his reasoning! He thought that the probability of winning would be 2/3 in both games. The incorrect reasoning let him to believe that betting at even odds would be a winning proposition. So he thought. Though his reasoning was wrong in the first game, he was lucky that the winning odds were still better than even. For the second game, he learned the hard way – through simulation with real money!

There are two issues involved here. One is obviously the flawed reasoning in probability on the part of de Méré. The second is calculation. de Méré and his contemporaries would have a hard time making the calculation even if they were able to reason correctly. They did not have the advantage of calculators and other electronic devices that are widely available to us. For example, the following shows the calculation of the winning probabilities for both games.

$\displaystyle P(\text{at least one six})=1 - \biggl( \frac{5}{6} \biggr)^4=0.518$

$\displaystyle P(\text{at least one double six})=1 - \biggl( \frac{35}{36} \biggr)^{24}=0.491$

It is possible to calculate 5/6 raised to 4. Raising 35/36 to 24 would be very tedious and error prone. Any one with a hand held calculator with a key for $\displaystyle y^x$ (raising y to x). For de Méré and his contemporaries, this calculation would probably have to done by experts.

The main stumbling block of course would be the inability to reason correctly with odds and probability. We have the benefits of the probability tools bequeathed by Pascal, Fermat and others. Learning the basic tool kit in probability is essential for anyone who deal with uncertainty.

One more comment about what Chevalier de Méré could have done (if expert mathematical help was not available). He could have performed simulation (the kind that does not involve real money). Simply roll a pair of fair dice a number of times and count how many times he wins.

He would soon find out that he would not win 2/3 of the time. He would not even win 51% of of the time. It would be more likely that he wins 49% of the time. Simulation, if done properly, does not lie. We performed one simulation of rolling a pair of dice 100,000 times (in Excel). Only 49,211 of the iterations have “at least one double six.” Without software, simulating 100,000 times may not be realistic. But Chevalier de Méré could simulate the experiment 100 times or even 1,000 times (if he hired someone to help).

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$\copyright$ 2017 – Dan Ma