Outreach module (grades 9-12)
Part 2: Investigation plan
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 is a guided walkthrough for how to approach the problem before using the lab.
There are several different questions hiding inside this problem. Let's start by listing out what problems we want to answer.
We will approach them by first thinking through some examples.
It is often helpful to start by thinking about simple and extreme examples and then trying to draw lessons from them.
Suppose you estimate that the total amount of money in the world is on the order of \(100\) trillion dollars. If you open your envelope and see more than half of that amount, what should you do?
You should stay. The other envelope cannot possibly contain double what you saw, so your envelope cannot be the smaller one.
This conclusion does not depend on how \(X\) is generated. The observation itself forces the answer.
Suppose \(X\) is chosen uniformly from \(\{1,2\}\). Then the envelope pair is either \(\{1,2\}\) or \(\{2,4\}\). What should you do if you observe \(A=1\)? What about \(A=2\)? What about \(A=4\)?
If you observe \(A=1\), then you must have the smaller amount, so you should switch. If you observe \(A=4\), then you must have the larger amount, so you should stay.
If you observe \(A=2\), the situation is less clear. You could be holding the larger amount from the pair \(\{1,2\}\), or the smaller amount from the pair \(\{2,4\}\).
Since \(X=1\) and \(X=2\) are equally likely, and in each case you are handed either envelope with probability \(\frac{1}{2}\), those two ways to observe \(2\) are equally likely. So after observing \(A=2\), switching and staying are equally good in this model.
Now compare the same observation in two different models. In Model A, \(X=1\) every time, so the envelopes are \(\{1,2\}\). In Model B, \(X=2\) every time, so the envelopes are \(\{2,4\}\). In both models, suppose you observe \(A=2\). What should you do?
In Model A, observing \(2\) means you have the larger amount, so you should stay. In Model B, observing \(2\) means you have the smaller amount, so you should switch.
So the same observed value can lead to opposite decisions in different models.
From these examples, there are already a few questions we can answer.
Yes. In Example 1, observing more than half the money in the world forces you to stay. In that situation, it does not matter how \(X\) was generated.
Yes. In Example 2, when \(X\) is uniform on \(\{1,2\}\) and you observe \(A=2\), the two possibilities are equally likely, so switching and staying are equally good.
Yes. Example 3 shows that the same observed value \(A=2\) can tell you to stay in one model and switch in another. So sometimes knowing how \(X\) is generated really does matter.
No. Example 1 already shows that there are observations where you definitely should not switch. So Argument A cannot be right exactly as stated. Something in that reasoning has to go wrong.
These are the ideas to keep in mind as you move forward.
Before moving on, pause and think about a few follow-up questions.
Write down a short hypothesis about these questions and why you believe it.
On the next page, we will test these questions numerically in the lab.