A No-Math Resolution to the Two Envelopes Paradox

You’re presented with two sealed envelopes containing an unknown sum of money — except you’re told that one envelope contains twice as much as the other.

You are allowed to keep only one envelope.

With one in hand, Envelope A, say, you wonder how much is in the other, Envelope B. Perhaps it contains twice the one in your hand. But it might also contain half as much.

Then it dawns on you: double or nothing is already a fair bet — double or half is way better!

To make sure, you whip out the calculator app on your phone.

You guess what’s in Envelope A — $20, say. Then, there’s a 50% chance of Envelope B hiding $10 and a 50% chance of it hiding $40.

You type in (0.5 × 10) + (0.5 × 40), and the calculator returns 25. Your initial instinct was right — you should expect to gain $5 if you switch!

Envelope B is the winner!

But wait — if you’d started with Envelope B, you would make the exact same argument to switch to Envelope A. That’s obviously contradictory.

Welcome to The Two Envelopes Paradox.

Wait, hold on…

Before we continue, I just want to make sure it’s clear why switching is illogical.

Imagine a friend is playing with you, and they get the envelope you didn’t choose. For example, you have Envelope A, and they have Envelope B.

You both run through the same switching argument in your heads and simultaneously conclude that switching is preferable.

You can’t both be right — either you have the smaller amount, or they do.

This is all very abstract, I agree. But seeing is believing, so let’s just play the game!

Try it yourself

Curious to see what happens when you actually try it? I’ve built an interactive simulator that lets you experiment with how the envelope amounts are chosen. You can explore several options:

• Bounded: The amount in the smaller envelope is selected uniformly from a fixed range.
• Upper Bounded: Only a maximum is set — the smaller amount is drawn uniformly…