Ignoring the fact that the term "woman" might imply having a certain level of adult maturity, I would just like to point out that when I hear a baby cry I want to chuck it out the window. However my mother gets mad at me when I suggest that to the kid's parents, so maybe you have a point.

And while gender roles today could possibly be based off of biological differences from thousands of years ago, that doesn't mean that they still aren't viewed as confining to people today. Since (to my knowledge) there has never been a person raised in a completely gender-neutral environment, there's absolutely no basis for assuming that a person raised in a bubble unaware of their own sex would be better or worse at math or video games or chess or fixing cars or not crying or holding doors open or pulling chairs out or offering people their jackets or whatever it is we expect men to do, depending on their gender.