The events on Wall Street in the early 1990s, important as they were for the world economy, may be even more important for changing the fundamental understanding of probability and randomness. It was in the decade after 1654 when these ideas first entered intellectual and practical life and fused the concepts of frequency and degree of belief. It was in the decade of the 1990s that we finally figured out why that fusion makes sense.
Reading the 1654 letters between Fermat and Pascal, it's not obvious why they started an intellectual revolution. Probability is never mentioned, nor is randomness. The two great mathematicians struggled painfully with problems any high school student can easily solve today. Each of them came up with a different answer to the question of how to divide the stake in an interrupted game. The key of the letters is when Pascal realizes, to his enthusiastically expressed joy, that the two methods give the same answer.
Consider two gamblers playing a series of games. The first one to win seven games gets the stake. One gambler has six wins and the other five when the game is interrupted. How should the stake be divided?
Fermat noted that if the game were to continue, the result must be decided after two more games are played. Letting W represent a win for the first player (the one with six wins) and L represent a loss, the two games could come out in four ways: WW, WL, LW, and LL. In practice, in the first two of ...