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. so about 50%/10 or a 5% chance.” And there we sit, smug as a bug on a rug. After pounding your head with statistics, you know not to divide, but use exponents.
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).
Let’s see why the paradox happens and how it works.
We’ve taught ourselves mathematics and statistics, but let’s not kid ourselves: it’s not natural.
In a room of just 23 people there’s a 50-50 chance of two people having the same birthday.
In a room of 75 there’s a 99.9% chance of two people matching.
Put down the calculator and pitchfork, I don’t speak heresy.
The birthday paradox is strange, counter-intuitive, and completely true.
Did you naturally infer the Rule of 72 when learning about interest rates? Understanding compound exponential growth with our linear brains is hard. Notice how much of the negative news is the result of acting without considering others.
I’m an optimist and have hope for mankind, but that’s a separate discussion :). The fact that we neglect the 10 times as many comparisons that don’t include us helps us see why the “paradox” can happen.
In a room of 23, do you think of the 22 comparisons where your birthday is being compared against someone else’s? Do you think of the 231 comparisons where someone who is not you is being checked against someone else who is not you? The question: What are the chances that two people share a birthday in a group of 23?