The Newton-Pepys Problem, Measures of Central Tendency, and Symmetry of a Binomial Distribution

In 1693, the future famous diarist Samuel Pepys (1633-1703) sent Isaac Newton (1643-1727) a letter inquiring about certain probabilities related to rolling dice. Newton, who had read Christiaan Huygens’s (1629-1695) treatise on probability but showed little interest in the subject in his own work, sent a lengthy response to Pepys within four days; and both parties exchanged two additional letters on the topic within one month’s time. One can only speculate on reasons for Newton’s response to Pepys on a gambling question, much less three responses within a short period of time. Pepys lived in London and Newton in Cambridge, but they would have known each other because Pepys was the president of the Royal Society from 1684-1686 when the first edition of Newton’s

Principia was being prepared for publication. It was Pepys’s name that appeared under the imprimatur on the title page of the first printing of Newton’s great work. Perhaps that association influenced Newton, perhaps Newton was cooperative because both he and Pepys enjoyed the patronage of the Montagu family, or perhaps Pepys was simply lucky enough to have caught the great man when he was not otherwise absorbed. Whatever Newton’s motivation, he answered the following question posed by Pepys (Tanner, 1925):
Which is most likely

A. At least one 6 from the roll of six dice;

B. At least two 6’s from the roll of twelve dice;

C. At least three 6’s from the roll of eighteen dice?