paradoxes - How do you explain this probability paradox?

Imagine there are two bags of money, and you are allowed to choose one. The probability that one of them contains $10^{n-1}$ dollars and the other contains $10^{n}$ dollars is $1/2^n$, $n\in\{1,2,3...\}$.

That is to say, there is $1/2$ probability that one of the two bags contains $\$1$ and the other contains $\$10$; $1/4$ probability that one of the two bags contains $\$10$ and the other contains $\$100$ , etc.

What's interesting is that, no matter which one you choose, you'll find that the other one is better. For example, if you open one bag, and find there are $\$10$ in there, then the probability of the other bag contains $\$1$ is $2/3$ and the probability of the other bag contains $\$100$ is $1/3$, and the expectation of that is $\$34$, which is better than $\$10$.

If the other one is definitely better regardless of how much you'll find in whichever one you choose, why isn't choosing the other one in the first place a better choice?