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The "birthday problem" is one of several examples (Shaughnessy 1977, 1992) that illustrate students' tendency to underestimate the probability of an event's occurring at least once. Moore and McCabe (1993, 297) give the real-life example of "at least one false positive AIDS test result" when all persons tested are actually free of AIDS. Lesser (1997) discusses the event that at least one of six lotto numbers drawn without replacement was also drawn in the previous drawing. Mathematical probability theory itself was triggered in the midseventeenth century by examining the probability of at least one "double six" in twenty-four rolls of two dice (Katz 1993, 411).
The birthday-problem question, posed by Richard von Mises in 1939, is typically stated as "How many people must be in a room before the probability that some share a birthday, ignoring the year and ignoring leap days, becomes at least 50 percent?" A less common version is "Since a 100 percent chance of a match exists with 366 peopleeven if the first 365 people in the room had different birthdays, the pigeonhole principle forces the 366th person to match one of them-how many people are needed for a 99 percent chance of at least one match?" Take a moment to guess answers to these two questions before reading further.
RESULTS OF A SPREADSHEET RECURSION APPROACH
Historically, textbooks that included the birthday problem (e.g., Lapin [1987]) set up a long product of probabilities, which is a computationally tedious procedure even with a calculator. Using a recursive approach makes the process more compact, and using a spreadsheet makes the recursion more accessible. As Neuwirth (1996, 253) states ". . . recursion-troublesome for students both in conventional programming and in classical algebraic notation-appears to be little problem for most students using the point-and-click tabular representation in spreadsheets. Referring to the cells above a current cell seems very natural." The appendix explains the main Excel commands used in this article. If we define
Ai = the event that a group of i people have distinct birthdays,
the probability Pr(A1) is trivially 1. If we visualize the kth person joining a group of k - 1 people, then the event Ak happens when both (1) the event Ak_l happens and (2) the kth person.
