Lactose intolerance can be mediated slightly by taking probiotics, but if you already have crap bacteria inhabiting most of your innards then there's not much of a chance of the good stuff sticking. Interestingly enough the big one, gluten intolerance can also be mediated by probiotics. Certain strains (can't recall specifically which ones) affect zonulin which controls the permeability of your intestinal walls.
As far as hereditary goes, the most important probiotic a human ever gets is in their first breast feeding. This is when the mother lays down the colonies that will start the kid on their symbiotic journey.
"You can demonstrate the purpose and limits of human digestion with a simple experiment: eat a steak with some whole corn kernels, and see what comes out the other end. It won’t be the steak."