I don't think the government made a mistake, at birth, you were male. You went to school as a male, played sports as a male, joined the Boy Scouts as a male, possibly joined the military as a male and worked as a male. You married your wife and had children as a male. You can't change history.
At some point in your life you realized that you would be more comfortable in life as a female so you made that change. You are what you are until you change it. Now you are a female.
