Maddin

"So when you are able to reject H0 you can assure with a XX% confidence that the alternative (H1) is true, i.e. the drug is effective"

Just to muck things up a bit - When you reject H0, you are really saying that if H0 is true then you have drawn a sample so contrary to H0 that it would only happen 100*alpha % of the time. If H1 is the "omnibus" alternative (i.e., the one that includes everything that's not in H0) then either a rare event has happened or H1 is true. At some point (usu alpha = 0.05) we conclude that H0 is false because events rarer than 5% just aren't morally likely or something (We use 0.05 because RA Fischer one time said that 0.05 is no less arbitrary than any other number so we might as well use it).

The reason you can't exactly say that there is some probability that H1 is true is that H1 isn't a random event so it doesn't have probabilities associated with it.

Unless you're a Bayesian. And then you can say that stuff.