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Two-Envelope Scenario Calculator — result sheet
Evaluate a transparent two-envelope switching scenario under explicit up/down ratios and a stated probability.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: E[switch]=x[q·r_up+(1−q)·r_down]; break-even q=(1−r_down)/(r_up−r_down).
The classic paradox is under-specified without a prior model. This page therefore makes the two possible ratios and their probability an explicit scenario instead of pretending that switching is always favorable.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Amount in held envelope — minimum 1.0E-6; maximum 1000000000000
- Larger-envelope ratio — minimum 1.000001; maximum 1000
- Smaller-envelope ratio — minimum 1.0E-6; maximum 0.999999
- Probability of the larger case — minimum 0; maximum 1
Worked example
| Input | Value |
|---|---|
| Amount in held envelope | 100 |
| Larger-envelope ratio | 2 |
| Smaller-envelope ratio | 0.5 |
| Probability of the larger case | 0.5 |
Expected switch value = 125; expected change = 25; break-even larger-case probability ≈ 33.33%.
Assumptions and limits
- The held amount is positive and the two ratios describe the only two modeled cases.
- The probability of the larger case is supplied rather than inferred from the observed amount.
- Expected value depends on the stated scenario and cannot resolve every version of the paradox.
- This is a probability model, not financial advice.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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