The Two Envelopes Problem
A recent thread in reddit about the two envelopes problem reminded me of how unintuitive probabilities can be. There is a fundamental flaw with how the original post worded the problem:
You and I both have envelopes filled with money. My envelope contains either double or half the amount of money that’s in yours. If you want, I’m going to let you switch envelopes. Should you stay, switch, or does it not matter?
Stop right there. Read that again. Does that make sense to you? See if you can set up the experiment in the real world. Grab two envelopes and some money. Take $100 and put them in an envelope, then, uh... what do you do with the other envelope? Put $50 in it? Put $200?
The original poster then went ahead and solved assuming probabilities of 1/2. But I'm not really choosing between two equally likely options. I don't know the probability distributions!
A better wording for the problem (or a different problem if you're pedantic) would be:
There are two envelopes. One contains double the amount of money than the other. You choose one of them, I take the other. Now I offer you a choice between keeping your original choice or switching envelopes with me. Is it to your advantage to switch?
The answer should be obvious, it doesn't matter.
October 7th, 2009 - 06:18
There actually isn’t any flaw in how I worded the problem. Watch: I fill one envelope with $10, another with $20, and then give you the $20 envelope. I then tell you that my envelope contains either double or half the amount of money that’s in yours. There, what’s the problem? Setting up the problem is easy.
The reason this problem is controversial is because it’s TRICKY, and people hate it when they are tricked. Here is another similarly tricky problem: A couple has two children, one of whom is a boy. What is the probability that the other child is also a boy? The answer is a counterintuitive 1/3 because the problem does not state WHICH child is a boy. If the problem had instead stated, “the first child is a boy,” then the answer would be the expected 1/2.
Of course, when you explain that to people, they get mad and say, “Well, you tricked me! It’s an unfair question. You didn’t say which child was the boy, so I assumed it was the first one.” Exactly, you cannot make unfounded assumptions. That’s what the riddle is trying to teach you.
Likewise, the two envelopes problem is a lesson on symmetry. It’s TRICKY because people don’t realize that BOTH envelopes contain either double or half what’s in the other. It’s a symmetrical statement. Even if you reason yourself into switching envelopes, once you’ve switched it then turns out that my envelope STILL contains either double or half what’s in your envelope, so by your reasoning you should switch again. Which is, of course, ridiculous.
People like to complain, “It’s an unfair question. I thought you meant that my envelope was filled first with a constant value of $100, and then yours contains either $200 or $50.” Of course, that’s not what the problem says, and people are making unfounded assumptions.
Also, the incorrect solution that you are talking about, solution 1 on my site, was included on purpose specifically because it’s wrong. I was turning the riddle into one of those, “Here’s a proof that 1 + 1 = 3, where is the error?” type of problems.