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Newton-Pepys Problem


NewtonPepysProblem

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 n or more sixes in a roll of 6n dice is given by

P_n=sum_(x=n)^(6n)(6n; x)(1/6)^x(5/6)^(6n-x)
(1)
=1-sum_(x=0)^(n-1)(6n; x)(1/6)^x(5/6)^(6n-x)
(2)
=(5^(5n))/(6^(6n))(6n; n)_2F^~_1(1,-5n;n+1;-1/5),
(3)

where _2F^~_1(a,b;c;x) is a regularized hypergeometric function. Values for n=1, 2, and 3 are given by

P_1=(31031)/(46656) approx 0.6651
(4)
P_2=(1346704211)/(2176782336) approx 0.6187
(5)
P_3=(15166600495229)/(25389989167104) approx 0.5973
(6)

(OEIS A143162 and A143163). Therefore, P_1 is the most likely, with P_n asymptotically approaching 1/2 as n->infty.

Zeilberger (1996) gave a sum expression for P_n as

 P_n=1-2sum_(m=0)^(n-1)((94500m^4+214830m^3+171573m^2+56243m+6250)(6m)!5^(5m+2))/((5m+5)!m!6^(6m+5)),
(7)

from which the monotonicity of P_n 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 A_n Has 6n Dyes in a Box, With Which He Has To Fling [at least] n Sixes, Then A_n Has An Easier Task Than A_(n+1), at Eaven Luck." Amer. Math. Monthly 103, 265, 1996.

Referenced on Wolfram|Alpha

Newton-Pepys Problem

Cite this as:

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

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