What is the birthday paradox and why does it make me special?

Despite being 365 possible birth dates, a group of 23 people is enough to have a 50 % chance that someone share their birthday. With 70 people it raises to 99.9 %. Why?

At least, I had a cake just for me at home [photo: A♥]

At least, I had a cake just for me at home [photo: A♥]

In a group of 23, there are actually 253 pairs to compare: the first person with all the rest (22 comparisons), the second with all the rest except the first—that has already been considered—(21 comparisons)… adding up to 253 comparisons (23 × 22 : 2). This approach alone makes that 50 % less paradoxical.

Ignoring the 29th of February, the other 365 days would be equally likely to be someone’s birth date. The probability of two people not sharing birthdays is 99.7 % (364 de days out of 365 don’t match, 364/365 = 0,997).

The chance that no comparison be positive comes from multiplying the probability of a negative by itself as many times as pairs exist: 0.997253 = 0.500. That is, there’s a 50 % probability that nobody share birthday, meaning the other 50 % corresponds to at least one pair sharing.

Note that this is the probability that any pair share. The probability for a specific person comes from 1 − (364/365)n, where n is the number of people. For 23 people, that’s 6.1 %.

Why am I rambling about this? Remaking the numbers for 28 people—the approximate amount of people in my school, high school and Translation and Interpreting university classes—, I’ve had a 7.4 % probability of sharing my birthday three times. And it happened every time.

Multiplying the three probabilities, we get that the probability of that happening to me three times was 0.04 % (0.0743)—which makes me quite special since it only happens to 1 in every 2493 people. Moreover, I’ve also shared my birthday out of class. But there’s no need for more calculations; my ego’s been feed enough.

By the way, today is our birthday.

AZAD, Kalid. Understanding the Birthday Paradox. Better Explained. [viewed June 2014]

Birthday problem. Wikipedia. [viewed June 2014]


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