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Two Envelopes Paradox

Adjust envelope amounts and assumptions to examine why the paradox arises and how Bayesian reasoning resolves it.

Amount you believe is in the smaller envelope (x). The other envelope contains 2x.

Keep Envelope

$50.00

Expected value if you do not switch

Switch (Classic)

$62.50

Classic reasoning: 0.5·2x + 0.5·(x/2)

Bayesian Expected Value

$62.50

Uses subjective upper bound assumption

Smaller envelope assumed: $50.00 • Larger envelope: $100.00
Switching offers a higher expected value under the classic assumption.

How to Use This Calculator

  1. Set an assumed amount for the smaller envelope to model the scenario.
  2. Choose a strategy: always switch, never switch, or Bayesian with an upper bound.
  3. If using Bayesian logic, provide a maximum possible envelope value.
  4. Review expected values and recommendations to understand the paradox.

Formula

E[switch] = 0.5 · 2x + 0.5 · (x / 2) = 1.25x

Bayesian: P(smaller | x) ∝ Prior(smaller) · P(observe x | smaller)

E[switch | x] = P(smaller | x) · 2x + P(larger | x) · x/2

The paradox arises because the naive expectation ignores that x depends on the envelope chosen. Bayesian analysis resolves the inconsistency by modeling prior beliefs about envelope amounts.

Full Description

The two envelopes paradox poses a decision problem with two indistinguishable envelopes containing $x and $2x. Classical reasoning suggests switching improves expected value, creating a paradox. However, this reasoning fails because it treats x as fixed while it depends on the chosen envelope.

Bayesian modeling introduces prior beliefs about possible amounts, which restores coherence: depending on those beliefs, switching may or may not increase expected value. This calculator allows you to experiment with both views.

Frequently Asked Questions

Why does the classic reasoning fail?

It assumes a 50/50 chance that x is the smaller amount regardless of observation. In reality, observing x provides information about which envelope you chose, breaking that symmetry.

Does switching ever make sense?

Yes, if your prior suggests x is more likely the smaller amount. Bayesian reasoning quantifies when switching adds expected value.

What priors resolve the paradox?

Any proper prior distribution with finite expectation yields consistent advice. Improper or unbounded priors recreate the paradoxical expectation.

Is this relevant beyond puzzles?

The paradox illustrates how improper use of expected value can mislead decisions in Bayesian statistics and economics.