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LETTERS TO THE EDITORS

Another Probability Angle

To the Editors:

Theodore Hill’s article “Knowing When to Stop” (March–April 2009) discusses the marriage problem, where one attempts to maximize the odds of selecting the best spouse from a group. One encounters candidates in random order and can either accept one, and stop the process, or reject that candidate permanently. The best strategy is to assess and reject the first fraction 1/e (~37 percent) of the candidates and then pick the next candidate whose attributes equal or exceed the best in the initial sample.

I expect many people, instead of pinning their hopes on an unlikely shot at the very best candidate, would rather aim for the best they could expect on average. The strategy for this is different and somewhat counterintuitive. Assessing and rejecting only the first 9 of 100 candidates drops the odds of picking the absolute best candidate from 37 percent to about 22 percent. However, the expected rank of the chosen candidate rises from 81 to 91. In general, rather than being constant, the optimum fraction to be initially assessed decreases as the total number of candidates increases. The number varies from 1 for a total number of candidates in the range 3 to 5 up to 9 for a total number in the range 91 to 109.

Craig Stephan
Ann Arbor, MI


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