The Two Envelopes Problem (Paradox)
This is the famous Two Envelopes or Envelope Switching problem. People call it a paradox, but is it really? Let’s see.
But first a math joke:
Q: Why didn’t the Romans find algebra very challenging? A: Because X was always 10
Now, here’s the envelope problem: I give you two identical envelopes, each of which contains a sum of money. I tell you one envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it, I give you the chance to take the other envelope instead. Should you keep the first envelope or switch?
The answer is obvious, right? It doesn’t matter. You have a 50% chance of having the larger or smaller amount, and if you switch you still have a 50% chance of ending up with the larger or smaller amount. So, as Bill Murray said in Meatballs, it just doesn’t matter.
But wait. Not so fast. Watch what happens if we work it out mathematically:
Let A be the amount in the envelope you picked.
The other envelope has either twice that amount or half that amount.
So the other envelope has either 2A or 1/2A.
If you keep your envelope, you’ll end up with A.
If you switch, you’ll end up with 2A half the time and 1/2A half the time.
So if you switch, on average you will end up with the average of 2A and 1/2A. If that average equals A – which is what you started with – then, indeed, it doesn’t matter whether you switch or not, since on average you’ll end up with A dollars whether you switch or not.
Let’s average 2A and 1/2A and see if we get A.
To average them, we need to add 2A and 1/2A and divide by 2.
To add them, we need the least common denominator, which is 2. So we convert 2A to 4A/2. We add 4A/2 to 1/2A and we get the sum of 5A/2. We divide 5A/2 by 2 to get the average. And we get 5A/4 or 5/4A.
Wait a second!!!!! 5/4A is greater than A. On average, you’ll end up with 5/4A instead of A if you switch. So you need to switch, right?
Try to figure out if this makes sense. Can it really be that, whichever envelope you initially choose, it’s always wise to switch to the other? Is that logical? Is this a paradox?
Click here for the solution.
