The Birthday Paradox Problem

World & Nature

In a group of 23 people, there’s roughly a 50/50 chance of two people sharing the same birthday.

Increase that group size to 75 people, and it becomes a near certainty.

How does that make sense though? Surely if you’re in a room with 22 other people, you can only make 22 comparisons…

The thing is, that’s only the comparisons for one person. All of those 23 people need to be compared with each other.

I’ll let someone else explain:

When all 23 birthdays are compared against each other, it makes for much more than 22 comparisons. How much more? Well, the first person has 22 comparisons to make, but the second person was already compared to the first person, so there are only 21 comparisons to make. The third person then has 20 comparisons, the fourth person has 19 and so on. If you add up all possible comparisons (22 + 21 + 20 + 19 + … +1) the sum is 253 comparisons, or combinations. Consequently, each group of 23 people involves 253 comparisons, or 253 chances for matching birthdays.

(From: Probability and the Birthday Paradox)

Ok, but how do you get to 50% from 365 days and 253 comparisons?

Well, first you divide 364 by 365 to get the probability that a person doesn’t have the same birthday as someone else, which gives you 99.726027%.

Every one of the 253 combinations has the same odds, 99.726027 percent, of not being a match. If you multiply 99.726027 percent by 99.726027 253 times, you’ll find there’s a 49.952 percent chance that all 253 comparisons contain no matches. Consequently, the odds that there is a birthday match in those 253 comparisons is 1 – 49.952 percent = 50.048 percent, or just over half!

If you want to test out the probability for yourself with different group sizes, you can simulate an experiment here: Understanding the Birthday Paradox.

(That one gives the probability at 50.05% for 23 people but I think it’s because they rounded to 99.7260% when doing the calculation.)

Either way… it’s about 50%!

Use this fact to impress your friends, get the girl (/boy) and become rich and famous.

Or (more likely) cause the recipient of your fun fact to unenthusiastically say: “hmm, that’s interesting.”