Newton–Pepys problem

The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice.^[1]

In 1693 Samuel Pepys and Isaac Newton corresponded over a problem posed by Pepys in relation to a wager he planned to make. The problem was:

Which of the following three propositions has the greatest chance of success?

A. Six fair dice are tossed independently and at least one “6” appears.

B. Twelve fair dice are tossed independently and at least two “6”s appear.

C. Eighteen fair dice are tossed independently and at least three “6”s appear.^[2]

Pepys initially thought that outcome C had the highest probability, but Newton correctly concluded that outcome A actually has the highest probability.

Solution

The probabilities of outcomes A, B and C are:^[1]

$P(A)=1-\left(\frac{5}{6}\right)^{6} = \frac{31031}{46656} \approx 0.6651\, ,$

$P(B)=1-\sum_{x=0}^1\binom{12}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{12-x} = \frac{1346704211}{2176782336} \approx 0.6187\, ,$

$P(C)=1-\sum_{x=0}^2\binom{18}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{18-x} = \frac{15166600495229}{25389989167104} \approx 0.5973\, .$

These results may be obtained by applying the binomial distribution (although Newton obtained them from first principles). In general, if P(N) is the probability of throwing at least n sixes with 6n dice, then:

$P(N)=1-\sum_{x=0}^{n-1}\binom{6n}{x}\left(\frac{1}{6}\right)^x\left(\frac{5}{6}\right)^{6n-x}\, .$

As n grows, P(N) decreases monotonically towards an asymptotic limit of 1/2.

Newton's explanation

Although Newton correctly calculated the odds of each bet, he provided a separate intuitive explanation to Pepys. He imagined that B and C toss their dice in groups of six, and said that A was most favorable because it required a 6 in only one toss, while B and C required a 6 in each of their tosses. This explanation assumes that a group does not produce more than one 6, so it does not actually correspond to the original problem.

References

1. ^ ^{a b} Weisstein, Eric W., "Newton-Pepys Problem" from MathWorld.
2. ^ Isaac Newton as a Probabilist, Stephen Stigler, University of Chicago