The Birthday Problem

Picture yourself at a party...

What are the chances that two guests share the same birthday?

Art byrawpixel and vector4stock

High collision probability

In a group of just 23 people, there's already a 50% chance that two share the same birthday. Kinda surprising, right? With 365 days in a year, you'd expect collisions to be rare. But as the group grows, the number of pairwise comparisons grows even faster. In a group of 80, probability approaches 100%. Graph below shows how quickly the probability skyrockets.

Formula

Here's some math. The probability that at least two people in a group of n share a birthday is

Equation showing product of decreasing choices

Since the first person has 365 choices, the second 364, the third 363, etc...

Equation for probability all birthdays are different

Thus,

Closed-form probability formula

Birthday Simulator

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Intuition

Why does the probability of collision grow so fast? With each new person, we have to compare their birthday to everyone else's When a 24th person joins, we need to make sure they are not born on the same day as the previous 23. And so, The larger the group, the more connections we have to consider. Mathematically, this can be represented with a complete graph. See how complex the graph gets as the number of people increases.

Equation showing product of decreasing choices

Comparisons Demo

Comparisons: 253
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Thank you

More stuff coming soon!