The solution: No, there is no reason to change your choice.
the hint that is given is misleading! In this puzzle, it is not straightforward to calculate the expectation as suggested in the hint.
So, just use your common sense, and note that the chance is and stays 50% that your first choice will be the best one.
There is no reason at all to change that choice at the moment you find out that the box contains $10...
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