What is the Two-Envelope Paradox?

Hello! I'm excited to talk about the super interesting "Two-Envelope Paradox" with you. It's definitely something that can get your head spinning, but once you understand it, everything clicks into place.

What is the Two-Envelope Paradox?

Imagine this scenario:

Someone hands you two identical envelopes and tells you:

One envelope contains twice the amount of money as the other.
You don't know the exact amounts, but it's either A dollars in one and 2A dollars in the other.

Now, you randomly pick one envelope. Before you open it, that person asks you: "Do you want to switch to the other envelope?"

How should you decide?

The Seemingly "No-Lose" Switching Strategy

Most people might initially think that switching or not switching makes no difference; it's a 50/50 chance either way.

However, let's try to think about it in a "mathematical" way, and this is where the paradox begins:

Suppose the envelope you're holding contains X dollars.
Then, the other envelope can only contain one of two amounts:
- It's 2X dollars (if you happened to pick the envelope with less money).
- Or it's X/2 dollars (if you happened to pick the envelope with more money).
Since you picked randomly, both possibilities seem equally likely, each with a 50% probability.

Alright, now let's calculate the expected value of the money you would get by switching:

Expected Value = (Probability of getting 2X × 2X) + (Probability of getting X/2 × X/2) Expected Value = (50% × 2X) + (50% × X/2) Expected Value = X + 0.25X Expected Value = 1.25X

Wow! According to this calculation, the expected value after switching is 1.25X, which is 25% more than the X you currently have!

Based on this calculation, the conclusion seems to be: You should always switch!

Here's the paradox: If you switch and get the other envelope, before you open it, I ask you the same question again: "Do you want to switch back?" Following the exact same logic, if the new envelope you're holding contains Y dollars, the expected value of switching back is 1.25Y, so you should switch back... This leads to an infinite loop of switching.

This is clearly absurd. So, where exactly is the flaw?

Where Exactly Does the Paradox Go Wrong?

The core trick in this paradox lies in the fact that the variable 'X' is ill-defined and subtly swapped in meaning.

In the calculation above, X at one point represents the amount in the "smaller" envelope, and at another, it represents the amount in the "larger" envelope. Yet, within the same formula, it's treated as a fixed, known base value. This is logically incorrect.

Let's Think from a Different Angle

Forget that confusing X. Let's start from the initial setup. Let the amounts in the two envelopes be A and 2A.

The envelope you're holding can only be one of two cases:

Case One: You picked the envelope with A dollars. If you switch, you'll get 2A dollars, a net gain of A dollars.
Case Two: You picked the envelope with 2A dollars. If you switch, you'll get A dollars, a net loss of A dollars.

Both of these cases have a 50% probability.

So, let's calculate the expected net gain from switching: Expected Net Gain = (Probability of Case One × Gain) + (Probability of Case Two × Loss) Expected Net Gain = (50% × +A) + (50% × -A) Expected Net Gain = 0.5A - 0.5A = 0

The conclusion is clear: In terms of mathematical expectation, switching and not switching yield the exact same outcome. You don't "gain" anything by switching.

So, why is the 1.25X calculation wrong?

Because in the formula E = 0.5 * (2X) + 0.5 * (X/2), the two Xs do not represent the same thing.

In the 2X term, X represents the money in the smaller envelope (i.e., A).
In the X/2 term, X represents the money in the larger envelope (i.e., 2A).

You cannot use an X that represents A and an X that represents 2A in the same formula and treat them as the same baseline value to calculate the expected return. It's like adding the weight of an apple to the price of an orange; the units are inconsistent, making it completely meaningless.

An Easy-to-Understand Example

Suppose you open your envelope and find 100 dollars inside.

Now you're reconsidering whether to switch. The other envelope could contain either 50 dollars or 200 dollars.

Possibility A: The pair of envelopes contains (50, 100). You got 100, and switching would make you lose 50.
Possibility B: The pair of envelopes contains (100, 200). You got 100, and switching would make you gain 100.

Whether you should switch or not entirely depends on which you believe is more probable: Possibility A or Possibility B.

If the person who gave you the envelopes is a poor student, the probability of them having a (100, 200) combination (totaling 300) might be low, making the (50, 100) combination more likely. In that case, you shouldn't switch.
If the person who gave you the envelopes is Bill Gates, a (100, 200) combination would be trivial for him, and even (1 million, 2 million) is possible. In contrast, a "small money" combination like (50, 100) might be less probable. In that case, you probably should switch.

See? Once you acquire more information (such as opening the envelope and seeing the actual amount, and understanding the background of the person setting up the game), the decision is no longer a simple 50/50. The paradox's faulty calculation, precisely, ignores these crucial "prior knowledge" factors from the real world.

In Summary

The Two-Envelope Paradox is a very classic logical trap. It's not that mathematics itself is flawed, but rather that the way the problem is constructed is misleading.

Core Flaw: The "1.25X" calculation incorrectly mixes the meanings of the variable X.
Correct Perspective: From an omniscient viewpoint, if the amounts are A and 2A, the expected gain from switching is zero.
Real-World Decision: Once you open the envelope and see the specific amount, your decision is no longer a purely mathematical one. Instead, it's based on your probabilistic judgment (Bayesian inference) about "how this amount came to be."

So, the next time someone tries to "test" you with this problem, you can tell them that the seemingly clever 1.25X calculation is nothing more than a neat logical magic trick. 😉