✉️ Two Envelopes Paradox Calculator
The two envelopes paradox is a famous probability puzzle: one envelope has twice the money of the other, and a naive calculation seems to always favor switching. Enter the amount in your envelope to see the numbers behind the paradox.
What Is the Two Envelopes Paradox?
You are shown two envelopes and told one contains twice as much money as the other. You pick one, and before opening it, you are offered the chance to switch to the other envelope. Suppose your envelope has amount A. A tempting argument says: the other envelope contains either 2A or A/2, each with ‘probability’ 1/2, so its expected value is 0.5(2A) + 0.5(A/2) = 1.25A — which is more than A, so you should always switch. This calculator computes exactly that naive figure so you can see the numbers involved.
Why the Naive Argument Is Flawed
The catch is that the same argument would apply no matter which envelope you opened and no matter what amount A you saw, which cannot be right for both envelopes at once — switching can’t be simultaneously better from either envelope’s point of view. The flaw is treating A as if it is simultaneously the ‘smaller’ amount with probability 1/2 and the ‘larger’ amount with probability 1/2, for every possible value of A. In reality, whether your envelope is the smaller or larger one depends on the actual (fixed but unknown) pair of amounts placed in the envelopes, and a proper analysis requires a genuine prior probability distribution over what those amounts could be.
When you specify a sensible prior (for example, a distribution over the smaller amount that is not uniform over all positive numbers, which is impossible anyway), the paradox resolves: for any specific observed amount A, there generally is a well-defined optimal decision, but it is not true that switching is always better regardless of A. This calculator is intended for probability and statistics education to illustrate a classic reasoning trap, not as financial advice.