Ever be at a large birthday party and discover that two of the attendees have the same birthday? No, it is not cosmic karma. It happens more often than one would think.
The math that proves it is not supernatural is not overly challenging.
1. When you meet a random person, what is the chance that he or she has the same birthday as you do?
Well, without any other information, their birthday could be any day of the year. (Here we ignore that there are variations in the number of births between the days of the year. And we ignore leap year, too. Doing so will not impact the final result much.)
So the chance of the same birthday as you is one chance in 365. 1/365 = .00274 – a little less than three-tenths of one percent! The chance of no match is 1 – .00274 = .99726. This is why the result is a “paradox”. We intuitively expect no match happening to be almost certain.
2. Yes, the chance of one no match with your birthday is quite large. The thing is that at a gathering there are many people you might match with. And to have no match over all, you must not match with any of them.
Say, for example you are at a party with 23 attendees. Then, for each of the other 22, the chance of no match is .99726. The probability to match with none of them is .99726 x .99726 x .99726 .. x .99726. That is to say .99726 multiplied by itself 21 times. Using the superscript notation of algebra, this can be represented as (.99726)22 — the base probability raised to the twenty-second power.
So, at a party of 23, your chance of not finding a match is .941423. (If you have Microsoft excel, you can get this result by typing “power(.99726, 22)” into a cell.)
.941423 is still a very large probability. What more are we missing?
3. What we are missing is that the problem is not finding the chance of you having no match at the party, but of anyone there not having a match. This difference is the root of the paradox.
With 23 people, there are (23 x 22)/2 = 253 pairings. Each attendee can be matched with any of the other 22, but we have to divide by 2 to compensate for each pairing showing up twice. (That is to say, Tom paired with Sally is the same as Sally paired with Tom).
It is this factor of 253 that drives down the probability of there being no matches between any of the two people at the party.
4. So, our final probability is not (.99726)22 but (.99726)253 – a bucketload of more comparisons! This produces the final result of .4995. At a party of 23, the chances are roughly 50-50 that one pair of attendees to have the same birthday. (Actually, this result is not quite correct. It does not properly take into account when there are three or more people with the same birthday)
For larger groups, the steps above produce results that plummet even faster. For 35 attendees the chance of no match is only 19.55%. For 50, it is a mere 3.47%!
This is explained in a slightly different fashion at: https://betterexplained.com/articles/understanding-the-birthday-paradox/
As a bonus, near the bottom of the post is a calculator you can play with. And it discusses the approximations used in getting to the answers.