Paradox

QUINE’S QUESTION MARK 353concede that there are paradoxes in which the surprisingconclusion only purports to be probable. Consider the followinginstance of the birthday paradox. Professor Statisticspredicts that two of his students share a birthday from thepremise that there are forty students in the class. At first,the professor’s conclusion merely seems rash. The paradoxemerges when Professor Statistics divulges his reasoning:“‘There are forty students’ gives my conclusion a probabilityof 89.1 percent. To see why, picture a calendar with 365 dayson it. Mark your birthday. A second student now marks hisbirthday. She has a probability 364/365 of marking anempty day. The third person to mark the calendar has a 363/365 chance of marking an empty day. The chance that Npeople manage to mark an empty day is 1 - (365 × (365 - 1)× (365 - 2) . . . × (365 - (N-1)/(365N). So when there are 23people there is a 50.7 percent chance of a shared birthday.When N = 40, the formula implies that the probability of ashared birthday is 89.1 percent.”But having gone through all this, further suppose ProfessorStatistics has been unlucky: none of the forty studentsshares a birthday. His paradoxical prediction turns out to befalse even though it was backed by a true premise and anappropriate rule of inference.What makes the professor’s prediction paradoxical isthe reasoning behind it. The reasoning does not need to beperfect. Like most purveyors of the birthday paradox,Professor Statistics skated over the fact that some yearshave more than 365 days. Nor does he consider travelerswho lost their birthdays while crossing the InternationalDate Line. (An old man could die without having anybirthdays.) The paradox survives because these omissionsare insignificant.