Newton-Pepys Problem

The Newton-Pepys problem asks which of three events involving rolls of independent fair dice has the highest probability. Samuel Pepys posed it to Isaac Newton in correspondence concerning a wager (Schell 1960). The three events are

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. Stigler (2006) gives a detailed historical analysis of the error.

For a positive integer , the probability of obtaining or more sixes in a roll of independent fair 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.