What is the Birthday Paradox?

Tue, 02 Apr 2024 01:20:50 +0900

Do you know the Birthday Paradox?

Let me tell you a slightly mysterious story. How many people do you think need to gather for the “probability of people having the same birthday” to become high?

For example, a year has 365 days, so when you’re told “if 23 people gather, the probability of someone sharing a birthday is over 50%”… it feels somewhat counterintuitive, right?

But this is actually over 50%.

Why does this happen?

This phenomenon is called the “Birthday Paradox”. Its name contains “paradox”, but there is a proper mathematical reason for it.

When the number of people is “n”, the probability that no one shares a birthday can be calculated with the following formula:

1

P(No one shares) = 365/365 × 364/365 × 363/365 × ... × (365 - n + 1)/365

By subtracting that from 1, you get the “probability of sharing a birthday with someone”.

Looking at the results…

Number of People	Probability of people having the same birthday
10 people	Approx. 11.7%
20 people	Approx. 41.1%
23 people	Approx. 50.7% (Focus here!)
30 people	Approx. 70.6%
70 people	A whopping approx. 99.9%!

In other words, with just 23 people, there is a more than half chance that someone will share a birthday. It seems like this could apply quite often in a school class or a workplace meeting, right?

Conclusion: The gap between intuition and mathematics is interesting

The “Birthday Paradox” is an interesting example where our intuition and actual mathematical probabilities diverge. Knowing this kind of story might make for lively small talk or a fun quiz!

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