One envelope has 10 times the money in the other money.

WLOG let \(x\) be the envelope in Cary’s hand. The money in \(y\), then, \(y = \frac{1}{2}\qty(\frac{1}{10}x)+\frac{1}{2}\qty (10x) = 0.05x+5x = 5.05x\). Wat.

Basically; regardless if Cary took the envelope \(x\) or \(y\), the *other* envelope is expected to have \(5\times\) more money. What.

## Why?

There’s a bug in this:

\begin{equation} y = \frac{1}{2}\qty(\frac{1}{10}x)+\frac{1}{2}\qty (10x) \end{equation}

is not true! There is a human ** PRIOR BELIEF**!! Its very unlikely that mykel/chris put 10000 dollars into an envelope; so each individual amount in an envelope has an exogenous probability of it happening!