Probability Paradoxes That Will Blow Your Mind: Monty Hall, Birthday Problem, and More

Why Probability Paradoxes Fascinate Us

The first videos on the YouTube channel were about paradoxes. And several more have appeared since. We all love paradoxes — especially in probability theory. They are easy to state, easy to visualize, but nearly impossible to truly wrap your head around. Math often contradicts intuition. That’s probability theory for you.

Here are some paradoxes that have already been covered:

The Monty Hall Paradox

You’ve chosen one of three doors, behind one of which is a prize. Should you switch your choice after the host opens an empty door? The math shows that switching actually increases your chances of winning — even though it feels completely illogical.

The St. Petersburg Paradox

How much would you pay to play a game with an infinitely growing jackpot that seems like it should make you rich? Yet intuition tells us the game is nowhere near as profitable as it appears.

The Two Envelopes Paradox

You’ve opened one envelope and found some money. Should you swap it for the second envelope, knowing it contains either twice as much or half as much? It seems like there’s always a reason to switch — but intuition leads you straight into a dead end.

Probability Paradoxes in Everyday Life: The Boy or Girl Problem

If you learn that a family has a boy, what are the odds that their second child is also a boy? The answer is not as straightforward as it seems, and the logic behind it may genuinely surprise you.

The Unexpected Hanging Paradox

A judge announces that an execution will take place on one day during the following week, but the date will be a surprise. When the condemned tries to reason out which day it will be, he concludes that the execution cannot happen at all — which, of course, makes it all the more unexpected.

The Birthday Problem: A Classic Probability Paradox

What is the probability that in a group of 23 people, two of them share the same birthday? It turns out the coincidence is far more likely than you would ever guess.

The Sleeping Beauty Paradox

Sleeping Beauty wakes up not knowing what day it is, and must decide: what is the probability that today is Monday? The exact same set of facts leads to two different — yet equally defensible — answers.

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Which paradox is your favorite?

P.S. My personal favorite episode is the Boy or Girl Paradox, with its experimental non-linear narrative format — which, combined with a deliberately tangled explanation, makes it nearly incomprehensible. But it’s a great watch.