1.1 Getting Our Feet Wet

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1.1 Getting Our Feet Wet

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The start of probability theory can essentially be pinpointed to a single moment in time. In 1654 French nobleman Chevalier de Méré wrote to prominent mathematician Blaise Pascal asking for advice on the general “problem of points” (among other issues in making bets in gambling). These problems go as follows

Two friends each lay down $100 in a friendly “best of seven” tennis game, say. But rain interrupts play after just four games with one person having won three games and the other just one. How should the $200 be divvied between the two players so as to properly reflect the likelihood of each winning?

Pascal shared his thoughts with Pierre de Fermat. It is with de Méré’s inquiry that probability theory was born as a subject to be developed and deeply studied on its own right.

Comment: Italian mathematician Girolamo Cardano (1501-1576) actually worked with ideas akin to classical probability theory before this but did not publish his work. And of course gambling games have been in existence for centuries and scholars have wondered about, analyzed, and computed likelihoods of certain results. But the first definitive analysis of “chance” began with the work of Pascal and Fermat.

LET’S GET OUR FEET WET

I invite you to analyze de Méré’s problem. Imagine, like Pascal and Fermat, you are seeing this challenge for the first time and do not have all the usual tools in mind we usually associate with handling probability problems (tree diagrams, probability rules, and so on).

How would you approach the problem described above? What beginning assumptions would you make? What solution would you personally offer for splitting the pot between the two players?

(Do please think about this yourself before reading on.)

SOME POSSIBLE SOLUTIONS

Here are four possible solutions to the challenge. Start by giving the two gentleman names, A (for Albert) and B (for Bilbert), say, and suppose A has won three games and B just one.

Possible Solution 1: Just give each player back their $100 and play another day.

Possible Objection: A will likely complain that he was on the verge of winning and should receive more than this amount.

Possible Solution 2: Give player A $150 and player B $50 to reflect the ratio of their wins thus far.

Possible Objection: A might argue that if he were to win the next game he’d be given the full $200 and player B $0. This is better than the $160 and $40 distribution the 4:1 win ratio would suggest. Ratios undercut what A deserves.

Possible Solution 3: (See also lesson 2.1.) Imagine that this scenario happens over a large number of days, say, 4000 days, with A possessing three wins and B just one. But imagine for each of these days they do continue to play to see what happens next.

Suppose A and B are equally strong players, each just as likely to win any particular game.

With this assumption we would expect A to win the fifth game played on 2000 of those 4000 days pocketing the $200 right away. For the remaining 2000 days B will win the fifth game bringing the score to three wins for A, two wins for B, thereby forcing another match.

Among those 2000 days of play we expect A to win the sixth game played on 1000 of those days and pocket the $200. For the remaining 1000 days, B will win, bringing the score to three wins each, thereby forcing yet another match.

Of those 1000 days, A will win the seventh match played on 500 of those days and pocket the $200. For the remaining 500 days, B wins and pockets the $200.

Thus A pockets $200 on 3500 out of 4000 days and B pockets $200 on 500 out of those 4000 days. This suggests splitting the pot in a 7:1 ratio, which corresponds to giving A $175 and B $25.

Possible Objection: A would argue that the evidence suggests that he beats player B 75% of the time, not just 50% of the time.

Possible Solution 4: (See also lesson 2.2.) Follow the same analysis as above, but work with the assumption that A wins three-quarters of all matches played. This leads to the split $193.75 for A and $6.25 for B.

Possible Objection: B claims that he was “laying low” for the first few games for a psychological advantage and that he actually wins 95% of all games played!

Challenge: Find yet another reasonable solution to de Méré’s challenge.