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
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. Statistician14, 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. Monthly103,
265, 1996.