My mom was left-handed; she grew up in the 50's & 60's and said she had a hard time because everything -- sports equipment, notebooks, you name it--was all made for right-handers back then. My dad always told me he "made sure" my sister and I used our right hands for everything from the time we were born onward, and told us to make sure we did the same for our own kids because it was "a right-hander's world". I have since learned that you can "teach yourself" to use either hand. Using your non-dominant hand feels unnatural at first, but it's just a matter of muscle memory. People who have hands amputated or paralyzed make the switch all the time.
As for my two sons -- one is strongly left-handed, and the other is uniformly ambidextrous. What are the odds?