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What was the “gambler’s ruin”?

The gambler’s ruin is an application of the law of total probability that was first proposed by Dutch mathematician and astronomer Christiaan Huygens (1629–1695), although many people before him, including astronomer Galileo Galilei (1564–1642), brought up the same probability problem, but phrased it differently. By 1656, Huygens wrote a draft version of Van Rekeningh in Spelen van Geluck, a treatise about fifteen pages long based on what he heard about the correspondence of French scientist and religious philosopher Blaise Pascal (1623–1662) and French mathematician Pierre de Fermat (1601–1665) the previous year. Of the fourteen problems he presents, the last five became known as the “gambler’s ruin.”

In particular, Huygens (and others) wanted to find the probability of a gambler’s ruin. A common way of expressing the idea is by a game that has two players, with the game giving a probability q of winning one dollar and a probability (1 - q) of losing one dollar. In the problem, if a player begins with 10 dollars and intends to play the game repeatedly until he either goes broke or increases his holdings to 20 dollars, the question asked is: “What is his probability of going broke?” The answer involves quite a bit of probability computation. (For more information about the gambler’s ruin, see “Recreational Math.”)

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