What is the Two Envelopes Problem? How should it be understood?

Alright, no problem! Let's explain this interesting paradox in a conversational style.

What is the Two-Envelope Paradox?

Imagine you're on a TV show. The host presents you with two identical envelopes and tells you:

One envelope contains twice the money of the other.
You don't know the exact amount, but those are the rules.

Now, you randomly pick one envelope, let's call it Envelope A. Before you open it, the host gives you one last chance: "Do you want to switch to the other envelope (Envelope B)?"

At this point, you might think, it doesn't matter if I switch or not; it's a 50/50 chance either way.

But then, a "clever" thought pops into your head, and you start reasoning like this:

Assume the envelope in my hand, Envelope A, contains X dollars.

Then, the money in Envelope B has a 50% chance of being twice that of A, which is 2X dollars.

At the same time, it also has a 50% chance of being half of A, which is X/2 dollars.

So, if I switch to Envelope B, what is the "expected value" (which can be understood as the average gain) of the money I could get?

Let's calculate: Expected value E = (50% chance * getting 2X) + (50% chance * getting X/2) = 0.5 * (2X) + 0.5 * (X/2) = X + 0.25X = 1.25X.

Wow! The expected value is 1.25X, which is more than the X I currently have! Therefore, I should switch!

This reasoning seems flawless. But the most perplexing part is that this conclusion is completely independent of how much money you have in your hand (what X is). This means that even if you switch to Envelope B, you can still apply the same logic to conclude that you "should switch back to Envelope A." This creates an infinite "switch-switch-switch" loop.

This is the Two-Envelope Paradox: a seemingly perfect mathematical reasoning that leads to a highly absurd conclusion in reality – "you should always switch."

How should we understand this paradox?

Does this paradox sound confusing? Don't worry, its problem lies in a very clever logical trap. That "clever" reasoning above is actually incorrect.

The core of the error is this: it conflates two entirely different scenarios and misuses the variable "X."

Let's slow down and look at the problem more clearly.

1. What does "X" actually represent?

In that flawed reasoning, "X" was treated as a fixed, known value. But in reality, the money in your hand (X) is inherently uncertain.

Let's look at how the game is set up from the host's perspective. When the host prepares the money, they don't first think of an "X" and then prepare a "2X" and an "X/2."

They prepare it like this:

First, determine a smaller amount, let's call it A dollars.
Then, put double that amount, which is 2A dollars, into the other envelope.

So, the money in these two envelopes from the beginning is always the pair (A, 2A).

Now, you make your choice. You have two possibilities:

Scenario 1: You picked the envelope containing A dollars. In this case, X = A in your hand. If you switch, you'll get 2A. You gain A dollars.
Scenario 2: You picked the envelope containing 2A dollars. In this case, X = 2A in your hand. If you switch, you'll get A. You lose A dollars.

Before you open the envelope, both of these scenarios have a 50% probability.

Therefore, the true expected gain of your "switching" action is: Expected Gain = (50% chance of gaining A) + (50% chance of losing A) = 0.5 * (+A) + 0.5 * (-A) = 0

The expected gain is 0! This means that, probabilistically, the long-term outcome of switching versus not switching is exactly the same. This perfectly aligns with our intuition.

2. Why is the 1.25X calculation wrong?

That classic flawed calculation E = 0.5 * (2X) + 0.5 * (X/2) = 1.25X is wrong because it forcibly mixes the two scenarios described above.

When it calculates 0.5 * (2X), it implicitly assumes you are holding the smaller amount (your X is actually A).
When it calculates 0.5 * (X/2), it implicitly assumes you are holding the larger amount (your X is actually 2A).

A single variable X cannot represent the smaller amount A at one moment and the larger amount 2A at another moment within the same formula; this is where the error lies. You cannot use a variable with an undefined meaning for calculations.

3. Does the situation change after opening the envelope?

This is where the paradox gets really interesting. Once you open the envelope, the situation changes!

Suppose you open the envelope and find 100 dollars inside.

Now, X = 100 is known information. At this point, the money in the other envelope can only be one of two possibilities:

It could be 200 dollars (if the original pair of amounts was (100, 200))
Or it could be 50 dollars (if the original pair of amounts was (50, 100))

At this point, whether you should switch depends on which pair of amounts—(100, 200) or (50, 100)—you believe the host was more likely to have chosen.

For example: If you think the host is poor and less likely to prepare 1000 dollars than 10 dollars, then when you see 10,000 dollars in your hand, you might strongly believe the other envelope contains 5,000 dollars, not 20,000 dollars. In this case, you wouldn't switch.
Conversely, if you see only 10 dollars in your hand, you might think the other envelope has a much higher chance of containing 20 dollars than 5 dollars. In this case, you would switch.

So, once you gain information (by seeing the money in the envelope), your decision is no longer a simple 50/50 matter; instead, it becomes a judgment based on this information and your assumptions about the host's behavior.

To summarize

Source of the Paradox: The Two-Envelope Paradox stems from a seemingly correct but actually flawed mathematical reasoning that leads to the absurd conclusion of "always switching."
Core Error: The flawed reasoning confuses the meaning of the variable "X" within a single formula, causing it to represent amounts from two different scenarios simultaneously.
Correct Understanding (before opening the envelope): Before opening the envelope, the expected gain from switching versus not switching is exactly equal, both being 0. So, it doesn't matter whether you switch.
Correct Understanding (after opening the envelope): Once you open the envelope and see the specific amount, you gain new information. At this point, your decision to switch or not depends on your judgment and assumptions about "how the host initially chose the amounts."

So, next time someone "tests" you with this problem, you can tell them: that calculation yielding 1.25X is a clever trick, playing a mathematical sleight of hand with variable confusion!