I was looking for said point where you were mistaken, Chakka, and can't find it -- been lost to the annals of time. I had in mind that it was something to do with infinite sequences having a 100% probability of every possible combination occurring, though that may not have been you. Even assuming it was, it's a minor point -- but we have to be wary of citing "evidence and reason" with 100% confidence regardless, as sometimes even the experts get it wrong.
As regards GM, I don't know enough about it but I think the real problem people have is with who is doing the Science (as well as the Science itself, to some extent). People don't trust big copmpanies becuase they have a tendency to put profit first. We see this anyway with Pharmaceuticals often skewing their published results, and should watch out for it equally in GM companies just in case.
And, with regards to MMR, the public was bombarded with scaremongering from all parts of the media, and the Government seemed to be caught off guard. We know now who was right, and could have done then, but the media just flooded the papers with "MMR is dangerous". I'm inclined to blame them rather than "stupid parents". We rely on the media to report our Science news (Which in itself is a big mistake, but...), and if they are reporting the "such-and-such is dangerous" story some people are going to believe it. There was also the idea of portraying Wakefield as a champion whistleblower, fighting bravely against the iron fist of the state and vested interests, that is the sort of Story we love, adding fuel to the fire... To be sure, some parents did make a horrible choice of back-street vaccinations, but I don't know if it was really fear of evidence and reason. It was just that the "wrong evidence" saw far more light than the "right" evidence.
I hope that in future, in some way, I'll be spending time passing on what I know about Science to the next generations, be that via teaching, or tutoring, or even, dare I say it, through journalism. One way or another, I'd see it as my responsibility to pass on the message to others, and if they don't accept that then I can't have presented the argument convincingly enough. This may not always be true -- sometimes people won't recognise the truth if it's dangled in front of their faces -- but it's a good philosophy to live by, rather than labelling people as stupid or afraid of reason.