Pascal's Dice Odds Law on Double Sixes

Dice odds never change. No matter what set of dice we hold, if the odds of a certain dice result is, say, 35 to 1, any way we throw the dice or any number of times we roll the dice, the odds will still be 35 to 1. Agree?

So, the chances for any dice result are the same in each roll. Dice odds will begin to work for us over a long period of play. That's when the law of averages takes effect. When we have thrown the die numerous times we would come to the point when the real odds equal the actual results. But if we have just thrown the die a few times we would see that the odds would seldom work and we could lose a lot from this gambling situation, waiting for the law of averages to work, until we lose opportunity to recover losses.

Now, when we talk of the odds of rolling a particular result within particular times of rolling the dice is a more sophisticated matter. This is different from the odds of results being the same for any dice roll. Let's take the case of sixes and double sixes wager. There used to be a good gambler who made sixes betting a career. The condition was that Antoine, Gambaud the gambler, would bet on rolling a number 6 using just 4 shoots of a die. Having 6 probable results of one die roll, an even-money play would better fair with 3 die shoots; Antoine had possible odds. But the other players soon had better ideas and decided to play the double sixes bet.

So Antoine agreed. He would roll two sixes using 24 dice throws. The 4 rolls he was used to doing he multiplied by 6, the total combinations possible with a second die, and got a total of 24 throws. Remember, two dice yield 36 results and a mere single two-sixes result. He lost. Then he consulted a brilliant mathematician, Blaise Pascal.

True dice odds, Pascal said, are those against winning a roll and then multiply the odds with 0.693, which is the co-log of a hyperbolic log of 2 wins. This is consistent with 35 multiplied by 0.693 which amounts to 24.255 or 24.3 dice rolls, considering an even-money play. So, to achieve a win Antoine needed 26 to 27 throws. He won.

So take it from Pascal. True dice odds come over a series of throws long enough for the law of average to work.