Newton–Pepys problem


The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice.
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:
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:
These results may be obtained by applying the binomial distribution. In general, if P is the probability of throwing at least n sixes with 6n dice, then:
As n grows, P decreases monotonically towards an asymptotic limit of 1/2.

Example in R

The solution outlined above can be implemented in R as follows:

for

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.

Generalizations

A natural generalization of the problem is to consider n non-necessarily fair dice, with p the probability that each die will select the 6 face when thrown. If r is the total number of dice selecting the 6 face, then is the probability of having at least k correct selections when throwing exactly n dice. Then the original Newton–Pepys problem can be generalized as follows:
Let be natural positive numbers s.t.. Is then not smaller than for all n, p, k?
Notice that, with this notation, the original Newton–Pepys problem reads as: is ?
As noticed in Rubin and Evans, there are no uniform answers to the generalized Newton–Pepys problem since answers depend on k, n and p. There are nonetheless some variations of the previous questions that admit uniform answers:
:
If are positive natural numbers, and, then.
If are positive natural numbers, and, then.
:
If are positive natural numbers, and then.