# Solve The Two Envelopes Fallacy

This is a puzzle proposed to me by
a viewer in the coments of one of my last videos.
Say you are given two envelopes
and told that both have money inside,
but one has 2 times a much money as the other.
You can keep one of them.
You pick one at random,
but just before you open it you decide to do a little calculation
to check whether this was the right choice
or whether you should swap.
You say the amount of money in the envelope you have is some unkown X.
Then, let's figure out what's the expected value is of the amount in the other envelope.
Well, it's either half X or 2X, and both are equally likely,
so the expectation value should be this.
But that's bigger than X.
As in, in expectation the other envelope has more money than the one you currently have
and so you should switch.
Obviously, this is wrong.
Both envelopes are equally likely to contain the higher amount.
The point of this puzzle isn't for you to prove that to me.
What I want you to do is explain why this reasoning went wrong.
See, this is what we call a Fallacy.
When you have reasoning that seems correct,
but gives you a false result.
As opposed to a Paradox, which has correct reasoning,
but leads to a seemingly impossible result, which is actually correct.
Anyway, I think I have one way of explaining what went wrong,
but I'd be interested in seeing what you guys say,
and see if you have a different solution to me.