The concept of a maximum likelihood estimate is illustrated using a discrete example. An urn contains different colored marbles. Marbles are selected one at a time at random with replacement until one marble has been selected twice. What is the maximum likelihood estimate of the number of marbles in the urn?

Contributed by: Marc Brodie (Wheeling Jesuit University)

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The number of marbles in the urn is a random integer between 1 and 11. This example was motivated by various websites (including the author's campus website) that show a different picture each time the site is accessed. The sequence of maximum likelihood estimates based on the number of different colors observed before a repeat is obtained is A111097 in The On-Line Encyclopedia of Integer Sequences.