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365×364×363×⋅⋅⋅× 343 = 0.4927. 36523 Thus, there is a slightly better than 50% chance for a match. To be sure that 22 would not work, we should calculate, in the same manner, the probability of duplicate birthdays in a group of 22 people. If we do this, we find the probability of finding no matches to be .5243 showing that, with 22 people, there is less than a 50% chance of a match. This problem is often confused with the following related problem: if you are in a room with 23 other people, what is the chance that at least one other person in the room has your birthday? The chance that any one person had a different birthday than yours is 364/365. Hence the probability that all 23 other people have a different birthday than you do is (364/365)23 = 0.939. Thus, while there is a better than 50% chance that two people in the room have the same birthday, there is only about a 6% chance that someone else has your birthday. We note that the birthday provides a nice point of departure for a study of coincidences in general. An excellent introduction to techniques in this area is given by Diaconis and Mosteller*. We have assumed that all birthdays are equally likely to occur. This is not quite true. Here is a graph the summarizing the frequency of birthdays in the United States (1978-1987) given by Thomas Nunnikhoven.** For each month, we have plotted the average daily frequency for days in that month. If the distribution for birthdays were uniform, then (ignoring leap years) each frequency would be 1/365 = 0.00274. 27 * Methods for studying coincidences," Journal of the American Statistical Association, 84, 853-861. ** A birthday problem solution for non-uniform birth frequencies, The American Statistician, Vol. 46, No. 4 pp. 270-273.PDF Image | USING LOTTERIES IN TEACHING A CHANCE COURSE
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