Exploring the Birthday Problem with Spreadsheets.

The "birthday problem" is one of several exam ples (Shaughnessy 1977,1992) that illustrate students' tendency to underestimate the prob ability 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 mid seventeenth 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 people even if the first 365 people in the room had differ ent birthdays, the pigeonhole principle forces the 366th person to match one of them?how many peo ple 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,