Beginnings
A game popular in Florence around 1600 rested on the total score from three ordinary dice. The scores of Three, when all dice scored one, and Eighteen when they all scored six, arose rarely, with most scores near the middle of the range. You should check that there are six different ways of scoring Nine (e.g. 6+2+1, 5+2+2, etc.), and also six ways of scoring Ten. It was commonly believed that this ‘ought’ to make totals of Nine or Ten equally frequent, but players noticed that, over a period of time, the total of Ten occurred appreciably more often than Nine. They asked Galileo for an explanation.
Galileo pointed out that their method of counting was flawed. Colour the dice as Red, Green, and Blue, and list the outcomes in that order. To score Nine from 3+3+3 requires all three dice to show the same value, and that can happen in one way only, (3,3,3). But the 5+2+2 combination could arise as (5,2,2), (2,5,2),
or (2,2,5), so this combination will tend to arise three times as often as the former; and 6+2+1 arises via (6,2,1), (6,1,2), (2,6,1), (2,1,6), (1,6,2), and (1,2,6), so this combination has six ways to occur. A valid approach to how often we can get the different totals takes this factor into account, and does indeed lead to more ways of obtaining Ten than Nine. The Florentine gamblers learned a vital lesson in probability – you must learn to count properly .
In the summer of 1654, Pascal (in Paris) and Fermat (in Toulouse) had an exchange of letters on the problem of points . Suppose Smith and Jones agree to play a series of contests, the victor being the first to win three games; unfortunately, fate intervenes, and the contest must end when Smith leads Jones by 2-1. How should the prize be split?
Such questions had been aired for at least 150 years without a satisfactory answer, but Pascal and Fermat independently found a recipe that, for any target score, and any score when the contest was abandoned, would divide the prize fairly between them. They took different approaches, but reached the same conclusion, and each showered praise on the other for his brilliance. For the specific problem stated, the split should be in the ratio 3:1, with Smith getting 3/4 of the prize, Jones 1/4.