arXiv:math/0701089v1 [math.ST] 3 Jan 2007 ntewso fagamble. a of advice wisdom sought letter the He Pepys’ on matters. scientific But concern begun. not press did the for great preparation Newton’s when period same Principia the 1686, Novem- the through 30, of 1684 ber President from Ad- London as of of served Society instead Secretary Royal had former would who a He Affairs as diary. miralty Pepys not that of would known of have Newton aware but life been 1660–1669, his have years of details the intimate over the covering pub- posthumously diary his lished for today (1633– known Pepys best clarification. is 1703) for pressed Pepys informa- as more offering tion then and briefly, answering question first the letters, probability. three in with problem responded a Newton posing Newton Isaac to ter ihohrpiaecrepnec nPps( Pepys Newton in correspondence on published private were bearing letters other have the with of we Some probability. all and 23, almost and are 16 December 1693, and 26 November on problem, sa etna Probabilist a Stigler as M. Stephen Newton Isaac ae 29) h etr eectdi textbook a in cited ( were Chrystal ( by letters Pepys The in 72–94). completely pages more and 181) 129–135; pages 2, Vol. 06 o.2,N.3 400–403 3, DOI: No. 21, Vol. 2006, Science Statistical [email protected] hcg,Ilni 03,UAe-mail: USA 60637, Chicago, Illinois of Chicago, University Statistics, of Statistics, Department of Professor Service Distinguished

tpe .Silri retDwt Burton Dewitt Ernest is Stigler M. Stephen c ttsia Science article Statistical original the the of by reprint published electronic an is This ern iesfo h rgnli aiainand pagination in detail. original typographic the from differs reprint nNvme 2 63 aulPpswoealet- a wrote Pepys Samuel 1693, 22, November On h he etr etnwoet ey nthis on Pepys to wrote Newton letters three The nttt fMteaia Statistics Mathematical of Institute 10.1214/088342306000000312 a rsne oteRylSceyadits and Society Royal the to presented was 1889 .PPS PROBLEM PEPYS’ 1. nttt fMteaia Statistics Mathematical of Institute , bu rbe novn ie etnsaayi sdiscus made. is he analysis error an Newton’s to dice. drawn is involving attention problem a about Abstract. ae53,weeh aePepys’ gave he where 563), page , 06 o.2,N.3 400–403 3, No. 21, Vol. 2006, 1876–1879 . 2006 , n19,IacNwo nwrdaqeyfo aulPepys Samuel from query a answered Newton Isaac 1693, In o.6 ae 177– pages 6, Vol. , 1926 o.1, Vol. , This . 1825 in , 1 otoe etal( Westfall footnote; ( ( Schell D. ( David rence ( Pedoe attention Dan in- public by commentary dependently wide with reprinted a known were to selections little when brought were were they they but until exercise, an as problem 5 ooflltrtr,wlho e ehdmade. had he bet a on welsh term, ( later Mosteller’s colorful 35) using would, the he and was correspondence announced A the ended fact Pepys most in probable, that most the Pepys re- was by after questioning him (C) convinced peated these Newton when of but third probable, the initially Pepys that correspondence, thought the in emerged it As appear. “6”s three least at appear. “6”s two least at appears. “6” one least success? of chance greatest correspon- the in emerged it as dence: paraphrase prob- the in state lem will I ambiguous, somewhat noticed, question. the revisit to matter, useful the seems about it interest- thought and Newton an New- how casts of writer on error portion light The other ing major wrong. any a is solution that or ton’s noted these have of to seems none ( But Gjertsen and con- 427–428). lottery; some state had a problem to the opened nection that Pepys with, letter excuse first the his to credence unwarranted including notice, briefer so- ( it Newton’s Sheynin accorded and Others posed Pepys lution. problem the cussed r,ntbyCanyadBlad( Bullard and Chaundy notably ers, 1965 .Sxfi ieaetse neednl n at and independently tossed are dice fair Six A. the has propositions three following the of Which Newton as was, statement original Pepys’ Since .Egte ardc r osdidpnetyand independently tossed are dice fair Eighteen C. .Tev ardc r osdidpnetyand independently tossed are dice fair Twelve B. ae ,3–5 n ai( Gani and 33–35) 6, pages , 1971 1960 1959 ,wodsisvl eeae tt a to it relegated dismissively who ), .Teeatosadsvrloth- several and authors These ). ; 1980 1962 ae 9–9) h gave who 498–499), pages , ae 2–2)adEmil and 125–129) pages , e and sed 1958 ae 34) Flo- 43–48), pages , 1960 1982 ,Mosteller ), 1986 aedis- have ) 1965 pages , page , 2 S. STIGLER

2. NEWTON’S SOLUTION is a byproduct of a proof that for any N and any p, the difference between the mean and median of a Newton stated the solution three times during the binomial distribution is strictly less than ln(2) < 0.7 correspondence: first he gave a simple logical reason (Hamza, 1995). So when the mean Np is an integer for concluding that A is the most probable, then he the two must agree, and this implies in particular reported a detailed exact enumeration of the chances that in all these cases, in each of the three cases, and finally he returned to the logical argument and gave it in more detail. 1 1 P (X Np) 2 and P (X Np) 2 , Newton’s exact enumeration was elegant and flaw- ≥ ≥ ≤ ≥ less; it is equivalent to the solution as might be pre- and so in each case P (X Np) exceeds 1/2 by a ≥ sented in an elementary class today. Newton worked fraction of the probability P (X = Np). In fact, in from first principles assuming no knowledge of the the cases Pepys considered we have to a fair ap- binomial distribution; we can now express what he proximation P (X Np) 1/2 + (0.4)P (X = Np). ≥ ≈ found by this calculation in terms of a random vari- The ranking Newton calculated then reflects the fact able X with a Binomial (N, p) distribution as fol- that the size of the modal probability for a binomial lows: distribution, P (X = Np), decreases as N increases and the distribution spreads out, p being held con- A. P (X 1) = 31031/46656 = 0.665 when N = 6 stant. Indeed, as De Moivre would find by the 1730s, and p = 1/6.≥ P (X = Np) is well approximated by B. P (X 2) = 1346704211/2176782336 = 0.619 1/ (2πNp(1 p)) 1.07/√N when p = 1/6. So in when N = 12≥ and p = 1/6. p particular, the− probabilities≈ in A, B, C are about C. Here Newton simply stated that, “In the third 1/2+(0.4)(1.07)/√N, an approximation that would case the value will be found still less.” give values 0.67, 0.62, 0.60, which agree with the In fact, exact values to two places. Chaundy and Bullard (1960) provide a cumbersome rigorous proof that P (X 3) = 60666401980916/101559956668416 ≥ this sequence is decreasing, in some generality. = 0.597 Note that this approximation depends crucially upon the probabilities P (X 1), P (X 2) and when N = 18 and p = 1/6, as another of Pepys’ P (X 3) of A, B, C being P (≥X Np) [i.e.≥ P (X correspondents (a Mr. George Tollet) found after ≥ ≥ ≥ much labor, while trying to duplicate Newton’s re- E(X))] for the three respective distributions, and sults (Pepys, 1926, Vol. 1, pages 92–94). the result depends upon this as well. Franklin B. Pepys had originally thought that C was the most Evans observed this sensitivity already in 1961, find- probable; Newton’s logical arguments and his care- ing, for example, that P (X 1 N = 6,p = 1/4) = 0.8220 < P (X 2 N = 12,p =≥ 1/4)| = 0.8416 (Evans, ful enumeration of chances pointed in the contrary ≥ | direction. But while the conclusion Newton reached 1961). That is, the ordering of A and B that New- is correct, only the enumeration stands up under ton found for fair dice can fail for weighted dice, scrutiny. To understand why, it will help to develop and indeed will tend to fail when p is sufficiently a heuristic understanding of why A is the most prob- greater than 1/6, even though they be tossed fairly able. and independently.

3. A HEURISTIC VIEW 4. NEWTON’S LOGICAL ARGUMENT Pepys’ problem amounts to a comparison of three In his first letter to Pepys on November 26, 1693, Binomial (N, p) distributions with p = 1/6, namely Newton had been content to give a short logical ar- those with N = 6, 12 and 18. He desired a ranking of gument for why the chance of A must be the largest. P (X Np) for the three cases. Now, in all Binomial He dissected the problem carefully, and made it clear distributions≥ where the mean Np is an integer, Np that the proposition required that in each case at is also the median of the distribution (and indeed least the given number of “6”s should be thrown. the mode as well). This is always true, surprisingly Newton then restated the question and gave an ap- even in cases like those under study here, where the parently clear argument as to why the chance for A distributions are quite skewed and asymmetric. This had to be the largest: ISAAC NEWTON AS A PROBABILIST 3

“What is the expectation or hope of A to least one “6” among the six dice], but James may throw every time one six at least with six often throw a six and yet win nothing, because he dyes? can never win upon one six alone. If Peter flings a “What is the expectation or hope of B to six (for instance) four times in eight throws, he must throw every time two sixes at least with certainly win four times, but James upon equal luck twelve dyes? may throw a six eight times in sixteen throws and “What is the expectation or hope of C to yet win nothing. For as the question in the wager is throw every time three sixes at least with stated, he wins not upon every single throw with a 18 dyes? six as Peter doth, but only upon every two throws “And whether has not B and C as great wherein he throws at least two sixes. And therefore if an expectation or hope to hit every time he flings but one six in the two first throws, and one what they throw for as A hath to hit his in the two next, and but one in the two next, and so what he throws for? on to sixteen throws, he wins nothing at all, though “If the question be thus stated, it appears he throws a six twice as often as Peter doth, and by by an easy computation that the expecta- consequence have equal luck with Peter upon the tion of A is greater than that of B or C; dyes.” (Pepys, 1926, Vol. 1, page 89; Schell, 1960) that is, the task of A is the easiest. And Here we can see more clearly how Newton was led the reason is because A has all the chances astray: Even though in the first letter he had care- of sixes on his dyes for his expectation, fully pointed out that “throwing a six” must be read but B and C have not all the chances on as “throwing at least one six,” here he confused the theirs. For when B throws a single six or C two statements. His argument might work if “ex- but one or two sixes, they miss of their ex- actly one six” were understood, but then it would pectations.” (Pepys, 1926, Vol. 1, 75–76; not correspond to the problem as he and Pepys had Schell, 1960) agreed it should be understood. Indeed, Peter will Newton’s conclusion was of course correct but the not necessarily register a gain with every “6”: if he argument is not. It is easy for us to see that it can- has two or more in the first “throw” of six dice, he not work because the argument applies equally well wins the same as with just one. Newton reduced the for weighted dice, and as we now know, the con- problem to single “throws” where each throw is a clusion fails if, for example, p is 1/4. Any correct Binomial (N = 6,p = 1/6), and he lost sight of the argument must explicitly use the fact that 1, 2, 3 multiplicity of outcomes that could lead to a win. are the expectations for A, B, C, and Newton’s does Many of Peter’s wins (those with at least two “6”s, not. His enumeration did do so, but A would equally which occurs in about 40% of the wins) would be well have “all the chances of sixes on his dyes” even wins for James as well. And in some of James’s wins if the chance of a “6” is 1/4. Newton’s proof refers (those with at least two “6”s in one-half of tosses only to the sample space and makes no use of the and none in the other half, about 28% of James’s probabilities of different outcomes other than that wins) Peter would not have done so well on “equal the dice are thrown independently, and so it must luck” (he would have won but half the time). Ev- fail. But Newton does casually use the word “expec- idently to make Newton’s argument correct would tations”; might he not have had something deeper in take as much work as an enumeration! mind? His subsequent correspondence confirms that he did not. 5. CONCLUSION In his third letter of December 23, 1693, Newton returned to this argument and expanded slightly on Newton’s logical argument failed, but modern prob- it. He personified the choices by naming the player abilists should admire the spirit of the attempt. It faced with bet A “Peter” and the player faced with was a simple appeal to dominance, a claim that all bet B “James.” He then considered a “throw” to be sequences of outcomes will favor Peter at least as six dice tossed at once, so then Peter was to make often as they will favor James. It had to fail because (at least) one “6” in a throw, while James was to the truth of the proposition depends upon the prob- make (at least) two “6”s in two throws. ability measure assigned to the sequences and the Newton then wrote, “As the wager is stated, Peter argument did not. But this was 1693, when proba- must win as often as he throws a six [i.e., makes at bility was in its infancy. 4 S. STIGLER

Why has apparently no one commented upon this David, F. N. (1962). Games, Gods and Gambling. Griffin, error before? There are several possible explana- London. tions, and no doubt each held for at least one reader. Evans, F. B. (1961). Pepys, Newton, and Bernoulli trials. Reader observations on recent discussions, in the series (1) The letters were read superficially, with no at- Questions and answers. Amer. Statist. 15 (1) 29. tempt to parse the somewhat archaic language of Gani, J. (1982). Newton on “a question touching ye different the logical proof, which after all points in the right odds upon certain given chances upon dice.” Math. Sci. 7 direction. (2) The language was puzzling and un- 61–66. MR0642167 clear to the reader (and Newton was not available Gjertsen, D. (1986). The Newton Handbook. Routledge and Kegan Paul, London. to ask), but it was accepted since he was, after all, Graves, R. P. (1889). Life of Sir William Rowan Hamilton Isaac Newton, and the calculation clearly showed 3. Hodges, Figgis, Dublin. Reprinted 1975 by Arno Press, he was sound on the important fundamentals. (3) New York. The reader may even have seen that it was not a Hamza, K. (1995). The smallest uniform upper bound on the satisfactory argument, but drew back from accus- distance between the mean and the median of the binomial and Poisson distributions. Statist. Probab. Lett. 23 21–25. ing Newton of error, particularly since he got the MR1333373 numbers right. Mosteller, F. (1965). Fifty Challenging Problems in Prob- In a sense the argument is more interesting be- ability with Solutions. Addison–Wesley, Reading, MA. cause it is wrong. Newton was thinking like a great MR0397810 probabilist—attempting a “eureka” proof that made Pedoe, D. (1958). The Gentle Art of Mathematics. Macmil- lan, New York. (Reprints the first two of Newton’s letters.) the issue clear in a flash. When successful, this is the MR0102468 highest form of mathematical art. That it failed is Pepys, S. (1825). Memoirs of Samuel Pepys, Esq. FRS 1, no embarrassment; a simple argument can be won- 2. Henry Colburn, London. (Reprints the first of Pepys’ derful, but it can also create an illusion of under- letters and two of Newton’s replies.) standing when the matter is, as here, deeper than Pepys, S. (1876–1879). Diary and Correspondence of Samuel Pepys Esq. F.R.S 1 6 it appears on the surface. If Newton fooled himself, , – . Bickers, London. (Reprints the first of Pepys’ letters and two of Newton’s replies.) he evidently took with him a succession of readers Pepys, S. (1926). Private Correspondence and Miscellaneous more than 250 years later. Yet even they should feel Papers of Samuel Pepys 1679–1703 in the Possession of J. no embarrassment. As Augustus De Morgan once Pepys Cockerell 1, 2. G. Bell and Sons, London. [This is wrote, “Everyone makes errors in probabilities, at the fullest reprinting. The portion of this correspondence times, and big ones.” (Graves, 1889, page 459) directly with Newton is fully reprinted in Turnbull (1961) 293–303.] Schell, E. D. (1960). Samuel Pepys, Isaac Newton, and REFERENCES probability. Published as part of the series Questions and answers. Amer. Statist. 14 (4) 27–30. [Schell’s article in- Chaundy, T. W. and Bullard, J. E. (1960). John Smith’s cludes a reprinting of the Newton-Pepys letters. Further problem. Mathematical Gazette 44 253–260. comments by readers appeared in Amer. Statist. 15 (1) Chrystal, G. (1889). Algebra; An Elementary Text-Book for 29–30.] the Higher Classes of Secondary Schools and for Colleges Sheynin, O. B. (1971). Newton and the classical theory of 2. Adam and Charles Black, Edinburgh. probability. Archive for History of Exact Sciences 7 217– David, F. N. (1959). Mr Newton, Mr Pepys & Dyse [sic]: A 243. historical note. Ann. Sci. 13 137–147. (This is the volume Turnbull, H. W., ed. (1961). The Correspondence of Isaac for the year 1957; this third issue, while nominally dated Newton 3: 1688–1694. Cambridge Univ. Press. MR0126329 September 1957, was published April 1959, as stated in the Westfall, R. S. (1980). Never at Rest: A Biography of Isaac volume Table of Contents.) Newton. Cambridge Univ. Press. MR0741027