The Two Envelope Paradox
You’re handed two sealed envelopes.
One has twice as much money as the other. You pick one. Let’s say it has $X.
Now comes the twisty part:
“Should you switch?”
You reason:
- There’s a 50% chance the other envelope has $2X
- A 50% chance it has $X/2
- Expected value = (0.5 × 2X) + (0.5 × X/2) = 1.25X
Whoa, more money? Always switch!
But wait—if that logic works no matter which envelope you pick…
shouldn’t you just keep switching forever?
🌀 Paradox activated.
What’s going on?
The problem hides in the assumption. You can’t just assign fixed probabilities to X without knowing the distribution of the amounts inside. You’re mixing up fixed values and variables.
It feels like free money, but without knowing how the envelopes were filled, the expected value math breaks down.
In short:
Infinite switching? Bad idea. Stick to your first love—uh, envelope.
Moral of the story
Sometimes in probability, reasoning too hard makes you lose your cash and your mind.
🎩💸✨