How to gamble if you must—the mathematics of optimal stopping

“Every decision is risky business. Selecting the best time to stop and act is crucial. When Microsoft prepares to introduce Word 2020, it must decide when to quit debugging and launch the product. When a hurricane veers toward Florida, the governor must call when it’s time to stop watching and start evacuating. Bad timing can be ruinous. Napoleon learned that the hard way after invading Russia. We face smaller-consequence stopping decisions all the time, when hunting for a better parking space, responding to a job offer or scheduling retirement.

The basic framework of all these problems is the same: A decision maker observes a process evolving in time that involves some randomness. Based only on what is known, he or she must make a decision on how to maximize reward or minimize cost. In some cases, little is known about what’s coming. In other cases, information is abundant. In either scenario, no one predicts the future with full certainty. Fortunately, the powers of probability sometimes improve the odds of making a good choice.

While much of mathematics has roots that reach back millennia to Euclid and even earlier thinkers, the history of probability is far shorter. And its lineage is, well, a lot less refined. Girolamo Cardano’s famed 1564 manuscript De Ludo Aleae, one of the earliest writings on probability and not published until a century after he wrote it, primarily analyzed dice games. Although Galileo and other 17th-century scientists contributed to this enterprise, many credit the mathematical foundations of probability to an exchange of letters in 1654 between two famous French mathematicians, Blaise Pascal and Pierre de Fermat. They too were concerned with odds and dice throws—for example, whether it is wise to bet even money that a pair of sixes will occur in 24 rolls of two fair dice. Some insisted it was, but the true probability of a double six in 24 rolls is about 49.1 percent…”,y.2009,no.2,content.true,page.1,css.print/issue.aspx