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The Birthday Paradox

See how exponential match can seem like magic!

Sometimes the most logical things don’t seem logical at all. Like brussels sprouts are good or in a room of 23 people there is a 50% chance that two of them will share a birthday. Bump that up to 75 people and you have a 99.9% chance of two people sharing a birthday.

How can this be? With 365 possible days to be born, how can the chances be so high for so few people?kids posing for photo

It’s a matter of perception. We are good at understanding ideas from our point of view. Math problems included.

Most people also have issues understanding non-linear math. Addition, subtraction, multiplication, and division make sense to us: they all act in the first person. They’re linear. We’ve been doing them since we were in elementary school. They’re just familiar operations.

But exponential math, like the birthday paradox, involves having to take a lot of factors into account. It isn’t solving for a single equation; it’s solving for multiple equations that are all linked together.

The reason the paradox works is because it’s not just you happening to match with one other person, it’s anyone in the room matching. It’s you times everyone else and the person next to you times everyone else minus their match with you and the person next to them times everyone else minus their match with you… You get the point.

Think of it as high-fives. It isn’t just you giving 22 high-fives. It’s 23 people each giving 22 high-fives. The math looks like this, 22x23/2=253. That give us 253 possible pairs. You divide by two because each high-five only counts once and you need two people to high-five.

So it turns out that the birthday paradox isn’t actually a paradox. It makes sense; there’s no fluke. The math is actually pretty simple. It just seems like a paradox because we often fail to see exponential results.


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