THE BIRTHDAY PARADOX

by Wesley Fager

My good friend John Rensch’s son says that a paradox is a partnership of two doctors. My paradox deals with birthdays. Let me start right off with the answer: In a room of fifty people --  it is almost certain that two will have the same birthday! Now that’s pretty astounding, if you ask me. Let’s prove it.

We will approach the problem by determining the probability that none in a given group have the same birthday and subtracting from one. Thus, for example, if it can be shown that the probability is .03 that no two people out of 50 have the same birthday--then the probability that two do is: 1 - .03 = .97 or 97%. The following assumptions are made: a year is 365 days and it is equally probable that an individual may be born on any day of the year. These assumptions are not entirely correct, but their effect on the probabilities is negligible. Consider two people. The likelihood that they are born on the same day is 1 out of 365 or 1/365; the probability they are not is 364/365. Consider three people: A, B, and C. The probability that A and B are born on different days is 364/365. Since A and B account for two of the 365 days, there are only 363 possible days left on which C can be born. Hence the probability that C is born on yet a different day is 363/365. The probability that all three have different birthdays is the product 364/365 X 363/365. (When tossing a coin, the probability of tossing a head is 1 out of 2 or 1/2. The probability of tossing three heads in a row is 1/2 X 1/2 X 1/2 = 1/8) The probability of four people having different birthdays is 364/365 X 363/365 X 362/365. And, in general, the probability that at least 2 out of n people in a given population have the same birthday is: 1 - (364/365 X 363/365 X . . . (365 - n + 1)/ 365).

The following table was developed using the above formula:

Number of People Probability at least two Have Same Birthday
5 0.027
10 0.117
15 0.253
18 0.347
20 0.411
23 0.507
25 0.569
27 0.627
30 0.706
35 0.814
40 0.891
50 0.970
60 0.9951
70 0.99916
80 0.99991
90 0.99999
100

So ladies, the next time you crowd 50 people into your living room for a tupperware party, offer 5 to I odds that at least two people have the same birthday. It’s a good bet!

(Article written for Data Link circa 1982.)