Two Envelopes Paradox

A fascinating puzzle in probability and decision theory. Enter an amount you see in one envelope, and we'll analyze the expected outcome if you switch.

Example Scenarios

See how the paradox works with different initial amounts.

Scenario 1: Finding $20

example

You open an envelope and find $20. Should you switch?

Amount: 20

Scenario 2: Finding $500

example

A high-stakes game where your envelope contains $500.

Amount: 500

Scenario 3: A Small Amount

example

What if the envelope only contains $4?

Amount: 4

Scenario 4: Large Sum

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Imagine finding $10,000 in the envelope. The logic still suggests switching.

Amount: 10000

Other Titles
Understanding the Two Envelopes Paradox: A Comprehensive Guide
Delve into the depths of one of probability theory's most counter-intuitive puzzles. This guide explores the setup, the flawed logic, and the accepted resolutions.

What is the Two Envelopes Paradox?

  • The Basic Setup
  • The 'Always Switch' Argument
  • Why It Feels Wrong
The Two Envelopes Paradox, also known as the exchange paradox, is a puzzle that exposes flaws in intuitive reasoning about probability. It involves a simple scenario: two indistinguishable envelopes, one containing a certain amount of money (A), and the other containing double that amount (2A). You are allowed to pick one envelope, and you can either keep the money inside or switch to the other envelope. The question is: what is the optimal strategy?
The 'Always Switch' Argument
Suppose you pick an envelope and find $X inside. The paradox arises from the following line of reasoning. The other envelope must contain either $2X or $X/2. Since you had no prior information, you assume there's a 50% chance of each case. You can then calculate the expected value of switching: E = 0.5 ($2X) + 0.5 ($X/2) = $1.25X. Since $1.25X is greater than your current $X, it seems you should always switch. But this reasoning can be applied symmetrically, regardless of the amount, which leads to a contradiction.

Step-by-Step Guide to Using the Calculator

  • Entering Your Value
  • Interpreting the Results
  • Using the Examples
Our calculator is designed to demonstrate the flawed logic that creates the paradox. It's an educational tool to help you visualize the problem.
1. Entering Your Value (X)
In the 'Amount in Your Envelope (X)' field, input any positive number. This represents the amount of money you've found in the envelope you hypothetically chose.
2. Interpreting the Results
After clicking 'Analyze Paradox,' the calculator will show you the expected value of the other envelope based on the standard, but flawed, paradoxical argument. It explicitly shows the formula used and the conclusion that one should always switch. Crucially, it also displays a warning explaining that this is a paradox and why the logic is not sound in a real-world scenario.

Common Misconceptions and Correct Methods

  • The Flaw of the Unknown Prior
  • The 'A' vs 'X' Confusion
  • Bayesian Resolution
The core of the paradox lies in a subtle mathematical mistake. The simple calculation of expected value isn't applicable in the way it's presented.
The Flaw: The Variables are Not the Same
The error is in the setup of the expected value calculation. Let the smaller amount be A. Your envelope X is a random variable, which can be either A or 2A. The amount in the other envelope, Y, depends on X. If X=A, then Y=2A. If X=2A, then Y=A. The calculation E(Y|X=x) = 1.25x is incorrect because the two 'X' values in the formula (2X and X/2) are conditioned on different underlying states of the world. The 'X' if you hold the smaller amount is not the same value as the 'X' if you hold the larger amount.
Resolution: Using a Proper Probability Distribution
A key resolution involves realizing that the amount of money, A, cannot be selected from a uniform probability distribution across all positive numbers (such a distribution does not exist). In any realistic scenario, there is a probability distribution p(a) for how the initial amount A was chosen. Once you define a proper prior distribution for A, you can use Bayesian reasoning. For a given observed amount X, you can calculate the posterior probability of whether you are holding A or 2A. This often leads to a conclusion where switching is not always the best strategy. For example, if you see an extremely large amount of money, it's more likely you're holding the 2A envelope.

Real-World Applications of Paradoxical Thinking

  • Stock Market and Investments
  • Decision Making Under Uncertainty
  • Scientific Research
While the Two Envelopes Paradox is a brain teaser, the principles it explores have serious implications for real-world decision-making.
Financial Decisions
In investing, one might be faced with a choice to stick with a known asset or switch to an unknown one with potentially higher returns. The paradox teaches us to be wary of arguments based on 'expected value' that don't account for the underlying probability distributions and potential hidden information.
Evaluating Information
The problem is a great lesson in critical thinking. It shows how a seemingly logical argument can be built on a faulty premise. This is crucial when evaluating proposals, news, or data that might be presented in a misleading way.