Newton-Pepys Problem

Samuel Pepys wrote Isaac Newton a long letter asking him to determine the probabilities for a set of dice rolls related to a wager he planned to make. Pepys asked which was more likely,

1. At least one six when six dice are rolled,

2. At least two sixes when 12 dice are rolled, or

3. At least three sixes when 18 dice are rolled.

Pepys originally believed that the last case was most likely, but Newton correctly opined that the first outcome has the highest probability. While Newton's conclusion was correct, his argument actually contained an error in logic. However, this fact was not noted until pointed out in the detailed and historically detailed analysis by Stigler (2006).

The probability of obtaining or more sixes in a roll of dice is given by

			(1)
			(2)
			(3)

where is a regularized hypergeometric function. Values for , 2, and 3 are given by

			(4)
			(5)
			(6)

(OEIS A143162 and A143163). Therefore, is the most likely, with asymptotically approaching 1/2 as .

Zeilberger (1996) gave a sum expression for as

(7)

from which the monotonicity of is self-evident by inspection.