Outreach module (grades 9-12)
Part 1: Problem setup
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?
This page introduces two plausible arguments that point in different directions. In Part 2, you will test those arguments by building examples and writing a hypothesis before using the lab.
The other envelope seems like it should be either half of A or double A. If those two outcomes were equally likely, the expected value after switching would be
\[ \frac{1}{2}\left(\frac{A}{2}\right) + \frac{1}{2}(2A) = \frac{A}{4} + A = \frac{5A}{4} = 1.25A. \]
Since \( \frac{5A}{4} > A \), this sounds like a strong reason to switch every time.
There is also a simple symmetry argument: because the envelopes were shuffled, before opening anything each envelope is equally likely to be the larger one. So one might conclude something stronger: it cannot possibly matter whether you switch or stay.
In that view, both rules must always have the same probability of ending with the larger amount:
\[ \mathbb{P}(\text{win by switching}) = \mathbb{P}(\text{win by staying}) = \frac{1}{2}. \]
Pause here before clicking ahead. Spend a few minutes thinking, and commit to a hypothesis you can test.
Write down your current prediction in one or two sentences.