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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

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

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

Zeilberger (1996) gave a sum expression for as

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

See also

de Méré's Problem

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References

Chaundy, T. W. and Bullard, J. E. "John Smith's Problem." Math. Gaz. 44, 253-260, 1960.David, F. N. "Mr Newton, Mr Pepys & Dyse [sic]: A Historical Note." Ann. Sci. 13, 137-147, 1959.Gani, J. "Newton on 'A Question Touching Ye Different Odds Upon Certain Given Chances Upon Dice."' Math. Sci. 7, 61-66, 1982.Mosteller, F. "Isaac Newton Helps Samuel Pepys." Problem 19 in Fifty Challenging Problems in Probability with Solutions. New York: Dover, pp. 19 and 33-35, 1987.Pedoe, D. The Gentle Art of Mathematics. New York: Macmillan, 1958.Schell, E. D. "Samuel Pepys, Isaac Newton, and Probability." Amer. Statistician 14, 27-30, Oct. 1960.Sheynin, O. B. "Newton and the Classical Theory of Probability." Archive Hist. Exact Sci. 7, 217-243, 1971.Sloane, N. J. A. Sequences A143162 and A143163 in "The On-Line Encyclopedia of Integer Sequences."Stigler, S. M. "Isaac Newton as a Probabilist." Stat. Sci. 21, 400-403, 2006.Varagnolo, D.; Schenato, L.; and Pillonetto, G. "A Variation of the Newton-Pepys Problem and Its Connections to Size-Estimation Problems." Stat. & Prob. Let. 83, 1472-1478, 2013.Westfall, R. S. Never at Rest: A Biography of Isaac Newton. Cambridge, England: Cambridge University Press, pp. 498-499, 1980.Zeilberger, D. "If Has Dyes in a Box, with Which He Has to Fling [at Least] Sixes, Then Has an Easier Task than , at Eaven Luck." Amer. Math. Monthly 103, 265, 1996.

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Newton-Pepys Problem

Cite this as:

Weisstein, Eric W. "Newton-Pepys Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Newton-PepysProblem.html

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