Understanding the Birthday Paradox
23 people. In a room of just 23 people there’s a 50-50 chance of at least two people having the same birthday. In a room of 75 there’s a 99.9% chance of at least two people matching.
Put down the calculator and pitchfork, I don’t speak heresy. The birthday paradox is strange, counter-intuitive, and completely true. It’s only a “paradox” because our brains can’t handle the compounding power of exponents. We expect probabilities to be linear and only consider the scenarios we’re involved in (both faulty assumptions, by the way).
Here’s an example: What’s the chance of getting 10 heads in a row when flipping coins? The untrained brain might think like this:
“Well, getting one head is a 50% chance. Getting two heads is twice as hard, so a 25% chance. Getting ten heads is probably 10 times harder… so about 50%/10 or a 5% chance.”
After pounding your head with statistics, you know not to divide, but use exponents. The chance of 10 heads is not .5/10 but $.5^10$, or about .001.
