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 to Pepys by a school teacher named John Smith.
[2] The problem was: Which of the following three propositions has the greatest chance of success?
Pepys initially thought that outcome C had the highest probability, but Newton correctly concluded that outcome A actually has the highest probability.
The probabilities of outcomes A, B and C are:[1] 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: As n grows, P(n) decreases monotonically towards an asymptotic limit of 1/2.
The solution outlined above can be implemented in R as follows: 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.
[3] 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 (notice that actually the number of faces of the dice and which face should be selected are irrelevant).
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
ν
ν
be natural positive numbers s.t.
ν
ν
ν
ν
ν
ν
Notice that, with this notation, the original Newton–Pepys problem reads as: is
As noticed in Rubin and Evans (1961), 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: (from Chaundy and Bullard (1960)):[4] If
are positive natural numbers, and
are positive natural numbers, and
(from Varagnolo, Pillonetto and Schenato (2013)):[5] If
ν
ν
are positive natural numbers, and