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Here’s the video explaining what’s wrong with the two envelope fallacy.
I read a lot of different creative solutions from you guys.
Here is one that is satisfying for me.
Again, we have 2 envelopes, and for concreteness we’ll say one has $1 and the other two.
I pick an envelop but don’t open it.
Remember the fallacy went like this: my envelope has some unknown amount X, but the other envelope
has either 2X or 1/2 X.
So in expectation, it has X plus a quarter X, which is more than X.
So I should swap.
Now here is my solution.
Let’s think of it in terms of the actually money assigned, and then it will be super
obvious what’s wrong.
Say that, unknown to me, my envelope contains $1, so X=$1.
Then what does the other contain?
It’s $2, right?
But my previous calculation was trying to say it’s either half a dollar or $2.
And similarly, if my X is actually $2, then the other is $1 dollar, never $4 dollars.
So claiming that the other envelope contains either 2X or half X is wrong.
What that’s saying is, regardless of X this is true about the other envelopes value.
What’s true is that you don’t know if X is 1 or 2 dollars.
So
the expectation value of the second envelope is the probability your X is $1 times the
amount in the other envelope in that case plus the probability your X is $2 plus the
amount in the other envelope in that case.
And you put the numbers in, and you get $1.5.
What you’d expect- half way between $1 and $2.
So you don’t bother swapping
Thank you for all your comments about this fallacy.
Lots of you saw what the problem was, but what I also really really liked is that, even
when many of you couldn’t figure it out, you still commented with some ideas you had
about it and some hunches.
That’s the point of all these puzzle videos I make- it’s to get you thinking.
Whether you get the answer or not is irrelevant.