Introduction
Probability theory, in its classic and comprehensible form, is believed to have been formalized by the French mathematician Blaise Pascal in the seventeenth century, among others. The story is that Pascal was given the task of calculating the winning bet in the event of an interrupted game of chance (Shafer 6; Goswami et al. 2527).
Discussion
The problem posed to Pascal involved dice: two players take part in the game — assuming for convenience that both have bet $100 — and the player who can win the greatest number of times out of seven rounds takes the whole bet. However, after four games, the players quarreled and decided to stop playing. At that, the first player was able to win three games, and the second only one; three games remained unfinished, and the current ratio of wins between the first and second players is three to one. The question is, what division of the total bet, $200, would be fair in this case?
Both of these games involve the theory of probability because the result of each roll of the die is random and unrelated to the previous ones, and it is impossible to determine in advance exactly which die face will fall next. Indeed, there is an option to return the bets to both players, but such an outcome may seem unfair to the first one who has already won three times as many games; it is not hard to understand.
The second option is to divide the bets in the ratio in which the wins were: that is, give $150 to the first player and $50 to the second player. However, even in this case, there may be objections since the first player’s fourth win could bring him $200 and the second player $0, which may also seem unfair. This problem was solved by Pascal using combinatorics, for instance, with the assumption that both players are equally strong and that the first player’s three wins in four rounds do not define that player as the stronger player (Shafer 11). Then the probability of either of them winning the following games would be 50%.
If the first player had a 50% probability of winning the fifth game and scored four points out of seven, he would take the maximum bet of $200. If the second player were to win game five, that would create a 3-to-2 win distribution. In this case, winning the sixth round would bring $200 to the first player, while losing would equalize the odds for a total bet: 3 to 3.
Conclusion
Finally, in the last seventh game, when the odds would have been the same, either player could have won a total bet. It follows that the second player could have taken the total bet with probability (1/2)3 = 1/8 = 0.125 (Tanton 9:00). That is, the distribution of bets after four games would be fair at a ratio of 7 to 1, which means the first player would have to pick up $175.
Works Cited
“Global Math Project: Garden Paths L4.” YouTube, uploaded by James Tanton. 2016. Web.
Goswami, Akhil, Gautam Choudhury, and Hemanta Kumar Sarmah. “Role of Fibonacci, Blaise Pascal, Pierre de Fermat and Abraham de Moivre in the Devolopment of Number Patterns and Probability: A Historical Search.” International Journal of Applied Engineering Research, vol. 14, no. 11, 2019, pp. 2527-2535.
Shafer, Glenn. “The Game-Theoretic Probability and Finance Project.” Working Paper, 53, 2015, pp. 1-25.