I ignored cases where the flop is Q22, or 4JJ. Yeah you have 2 pair, but that doesn't mean your hand is any good which is why straight probabilities aren't useful.

I outlined exactly how to calculate manually. You have to think about the different ways to make a hand and count them up.

If you're math minded there is interesting stuff here:
http://www.math.sfu.ca/~alspach/computations.html