Outreach module (grades 9-12)
Part 3: Envelope Problem Lab
Problem Setup
Two envelopes contain X dollars and 2X dollars, where X is a positive integer. The envelopes are shuffled, and one is handed to you at random. You open it and observe A dollars. Before payment, you may decide to keep your envelope or switch to the other one. You are paid whatever is in your final envelope. In this module, we call it a win if your final envelope has the larger amount. Should you switch if your goal is to end with the larger amount?
Use this lab to test the questions from the previous pages. Start with a hidden setup and compare always switch with always stay. Then try making your own decisions one round at a time, and see what changes when the setup is revealed.
In this lab, \(X\) is always a power of \(2\). This avoids giving away extra information from the parity of the number you observe.
In Mystery setup, the exact distribution is hidden. In Custom setup, you choose the highest power and the tilt. If the tilt is \(1\), the distribution is uniform. Larger tilt makes larger values more likely, and smaller tilt makes them less likely.
Each setup is determined by the highest power and the tilt. We write these as \(K\) and \(r\):
\[ X\in\{2^0,2^1,\dots,2^K\}, \qquad \mathbb{P}(X=2^i)\propto r^i. \]
So the highest power controls which powers of \(2\) are possible, and the tilt controls whether larger values are more likely or less likely.
First compare always switch and always stay in a mystery setup. Then use single round mode to decide case-by-case from the observed value \(A\).
After that, reveal the hidden setup or choose your own. The goal is to see whether knowing how \(X\) is generated changes what you can do.
Record one setup at a time. Here, win rate means the fraction of rounds where your final envelope had the larger amount.
| Setup stage | Rule | Rounds | Win rate |
|---|---|---|---|
| Mystery | Always switch | 0 | -- |
| Mystery | Always stay | 0 | -- |
| Mystery | Single round decisions | 0 | -- |
| Revealed | Always switch | 0 | -- |
| Revealed | Always stay | 0 | -- |
| Revealed | Single round decisions | 0 | -- |
Observed amount (\(A\)): \(A = \text{--}\)
Open an envelope to see this round's summary.
Setup
Current setup
Mystery setup is active.
Change the hidden setup or reveal its highest power and tilt.
Highest power means \(X\) can be \(1, 2, 4, \dots, 2^{\text{highest power}}\). Tilt \(=1\) is uniform.
Play
Open one envelope, then decide whether to switch or stay for that value.