Probability 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 to Pepys by a school teacher named John Smith. The problem was:
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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:
formula_1
formula_2
formula_3
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 6"n" dice, then:
formula_4
As "n" grows, P("n") decreases monotonically towards an asymptotic limit of 1/2.
Example in R.
The solution outlined above can be implemented in R as follows:
for (s in 1:3) { # looking for s = 1, 2 or 3 sixes
 n = 6*s # ... in n = 6, 12 or 18 dice
 q = pbinom(s-1, n, 1/6) # q = Prob( ;s sixes in n dice )
 cat("Probability of at least", s, "six in", n, "fair dice:", 1-q, "\n")
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 (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 formula_5 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 formula_6 be natural positive numbers s.t. formula_7. Is then formula_8 not smaller than formula_9 for all "n, p, k"?
Notice that, with this notation, the original Newton–Pepys problem reads as: is formula_10?
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)):
If formula_11 are positive natural numbers, and formula_12, then formula_13.
If formula_14 are positive natural numbers, and formula_15, then formula_16.
(from Varagnolo, Pillonetto and Schenato (2013)):
If formula_17 are positive natural numbers, and formula_18 then formula_19.
References.
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