I was about today years old when I learned of the two envelopes problem during one of my not-so-unusual attempts to do a breadth-first-search of the entirety of Wikipedia. Below is a summary of the relevant parts of the relevant article. (For your convenience, I omitted some irrelevant details in the "switching argument".)

Problem

A person is given two indistinguishable envelopes, each of which contains a sum of money. One envelope contains twice as much as the other. The person may pick one envelope and keep whatever amount it contains. They pick one envelope at random but before they open it they are given the chance to take the other envelope instead.

The switching argument

Now suppose the person reasons as follows:

1. Denote by A the amount in the player's selected envelope.
2. The probability that A is the smaller amount is 1/2, and that it is the larger amount is also 1/2.
3. The other envelope may contain either 2A or A/2.
4. ...

The puzzle

The puzzle is to find the flaw in the line of reasoning in the switching argument. This includes determining exactly why and under what conditions that step is not correct, to be sure not to make this mistake in a situation where the misstep may not be so obvious. In short, the problem is to solve the paradox. The puzzle is not solved by finding another way to calculate the probabilities that does not lead to a contradiction.

The Wikipedia article currently has 37 citations and describes a number of proposed solutions to the problem. I would assume that there are some insights in each of the described proposals, but I felt that reading them would most likely distract me with false leads. So, I'm not entirely sure what others have said about the problem. However, my solution is 101 words long. So, if I have to read 1,001 words in order to figure out whether someone else has solved it, then, in a certain sense, they haven't "solved" it. Maybe I haven't either. Maybe there is a 17-syllable haiku that could serve as a solution. But the compression algorithm underlying my uses of language is not, I suspect, haiku-complete.

Preface

The apparent existence of a problem here is a consequence of the way in which we are attempting to explain the apparent existence of a problem. Therefore, we could call it an illusory problem. However, such a label is misleading because even illusory problems can rather quickly turn into problems that are very real when people around us don't understand that the problems are illusory. (If you haven't encountered such a phenomenon in real life, I can assure you that it is by no means a purely hypothetical possibility.) So, our ability to succinctly explain the nature of the unreality of these illusory problems is a skill that can have significant real-world consequences. In other words, contriving such problems and discussing them is not at all a waste of time. If we can explain even one really well, then we'll probably find an anti-pattern that can crop up in multiple scenarios.

My Solution

Statement (2) begins, "The probability that $A$ is...". That's an interesting way to begin that statement, given the fact the variable $A$ is not a random variable. It was introduced to refer to an already-determined value. We don't know which value that is. However, there are three facts that we do know:

• There is one specific value that $A$ has with a probability of 100%.
• The probability of $A$ having any other value is 0%.
• There are no values whatsoever that $A$ has with a probability of 50%.

In light of the third fact, we see that statement (2) is false.

Further Analysis

In this scenario, a random variable would indeed need to be introduced in order to model the intuitive, common-sense reasoning that we would use in everyday life. However, we have to introduce the random variable into the problem statement at the point where common sense can still prevail over nonsense.^[1]

The fact that we don't know which value it is that $A$ has with a probability of 100% is indeed quite inconvenient. In fact, it means that the variable $A$ is almost surely entirely useless to us for any purpose other than allowing us an opportunity to explain its uselessness. But that was a state of affairs that we created for ourselves. There's no one else to blame. We have to accept responsibility for our own choices and not to confuse them with the choices made by other people, real or hypothetical.^[2]

Anti-pattern: Confusing the ignorance that we're trying to model with our own ignorance of how to specify that model or what we've actually specified.

More generally, the poor choices that we've made about how we describe or model problems cannot be overcome simply by adding more probability theory into our reasoning wherever we seem to get stuck.

If you want to know how this could all seem relatively obvious to me (which is not to say that I didn't spend a couple of hours trying to put this explanation into words), I will credit the fact that I spent much of my time in my Ph.D. program using automated proof assistants to check my proofs. If you know anything about such software, then you know that those tools don't cut you much slack. You learn to say what you mean and mean what you say, even if it takes you five times as long to say it. In the process, you also learn something about when and how a person can get fooled.

1. ^{^}

   Then again, who's to say what "common sense" really is? According to Principia Discordia, "Common sense tells us the earth is flat."

2. ^{^}

   I think Jimmy Buffett might have been trying to say something about that in one of his songs:

   Wasting away again in Margaritaville
   Searching for my lost shaker of salt
   Some people claim that there's a woman to blame
   But I know, it's my own damn fault