Envelope Trade I

There are two envelopes before you. You know that one of them contains $€2x$, while the other contains $€x$. However, you do not know what $x$ is. You take one envelope at random, observe the money inside, and have the option to switch to the other envelope. Should you switch the envelope? If any, which is the first incorrect line in the following argument? Answer with the number of the incorrect step. If the argument is correct, answer $-1$.

• (1) Let $M$ be the amount of money in the selected envelope.

• (2) The probability that $M$ is the smaller amount is $1/2$, while it is the larger amount with $1/2$.

• (3) The other envelope contains either $2M$ or $M/2$ inside.

• (4) If the envelope you selected is the smaller one, the other envelope contains $2M$.

• (5) If the envelope you selected is the larger one, the other envelope contains $M/2$.

• (6) The expected amount of money in the other envelope is $1/2 \cdot (M/2) + 1/2 \cdot 2M = 5M/4$

• (7) Since the expected value by swapping $(5M/4)$ is larger than the expected value of not swapping $(M)$, the optimal action is to swap.