Two Envelopes Paradox Calculator
Explore the famous probability paradox with simulation and calculation modes
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Calculation Mode
Enter the amount you observe in your chosen envelope
The Paradox Explained
The Setup
Two envelopes, one contains double the other
The Temptation
Other envelope seems to have 25% more on average
The Error
Mixing different scenarios in expected value calculation
Key Insights
Switching doesn't improve your chances
Both envelopes have the same expected value
The paradox arises from incorrect probability reasoning
Proper analysis considers all possible scenarios
Understanding the Two Envelopes Paradox
What is the Paradox?
The Two Envelopes Paradox presents a scenario where you're given two envelopes, one containing twice the amount of money as the other. After choosing one, you're offered the chance to switch. A flawed analysis suggests the other envelope always contains 25% more money on average.
The Flawed Logic
If your envelope contains X, the other contains either X/2 or 2X with equal probability. The expected value calculation E = 0.5 × (X/2) + 0.5 × (2X) = 1.25X suggests a 25% advantage for switching.
The Correct Analysis
The error lies in using the same variable X to represent different values in different scenarios. The correct approach considers two separate cases: when your envelope has the larger amount and when it has the smaller amount.
Correct Expected Value
E(other) = 0.5 × (L + S) = 0.5 × (2x + x) = 1.5x
Where L is the larger amount and S is the smaller amount.
Since both envelopes have this same expected value, switching provides no advantage.
Why This Matters
This paradox illustrates important concepts in probability theory and decision making under uncertainty. It shows how seemingly logical reasoning can lead to incorrect conclusions when probability calculations are not carefully considered. Understanding this paradox helps develop better intuition for expected value calculations and Bayesian reasoning.