Formant: Two envelope problem

Imagine a simple game. You are presented two envelopes and asked to choose one of them. Then you are shown that the envelope you have selected contains \( x \) monies. Then you are told that the other envelope contains either \( 10 x \) or \( \frac{x}{10} \) monies (with equal probabilities). Should you open the other envelope and take the monies in it? Or should you keep your original envelope?

The video below has participated in "Summer of Math Exposition" competition in 2021. It explains various ways to define and solve the two envelope problem and provides a valuable lesson on expected values.