No. 03
✉️Two Envelopes
The switch that looks profitable but isn't
The setup
- 1Two envelopes: one holds twice as much money as the other.
- 2You pick one. The 'expected value' argument says the other envelope averages 1.25× yours — so you should switch.
- 3But the same logic applies again after switching, and again... forever.
🤔 Intuition says: Expected value says switching beats staying by 25%. So always switch?
🎲 Play a round
Pick an envelope. You'll see what's inside — then find out whether switching would have helped.
Pick an envelope to begin.
📊 Always stay vs always switch
One round is luck. Let the server play thousands and compare an always-stay player with an always-switch player drawing from the same envelope pairs.
Why it's a paradox
The flawed step assumes every amount of money is equally likely, which would require an impossible infinite uniform distribution.
Once you account for a real prior over the amounts, the apparent 25% edge disappears.
Simulating many rounds confirms it: always-switch and always-stay earn the same on average.
💡 Key insight: You can't assume a flat prior over all amounts. The 'edge' was an artifact of bad bookkeeping.
Further reading on Wikipedia →