Two Envelopes Paradox Calculator

What Is the Two Envelopes Paradox?

Two envelopes contain money. One holds twice as much as the other. A flawed expected-value argument says the other envelope is always worth 25% more, no matter which one you hold. That cannot be true for a symmetric problem.

The error is treating one amount $X$ as both the smaller and larger value in the same formula:

$$E = \frac{1}{2}\cdot\frac{X}{2} + \frac{1}{2}\cdot 2X = \frac{5}{4}X$$

The correct approach uses the larger amount $L$ and smaller amount $S$ separately. See also the Monty Hall Problem Calculator and Expected Value Calculator.

Correct Expected Value

If one envelope has $x$ and the other $2x$, the expected value of either envelope is $(x + 2x)/2 = 1.5x$. Switching does not change your odds of getting the larger amount.