You can try this at home!
I have just tossed a coin ten times. I got three heads and seven tails. Should I assume that this coin has a probability of 0.7 for tails and 0.3 for heads? Let's see.
That sample of ten times was one of a hundred such samples that I recorded when tossing the same coin 1000 times. Taken as a whole, I had 503 heads and 497 tails, giving a probability (to two significant figures) of 0.50 for each possible outcome. What should I assume now? That the other 900 trials were flawed, or that my penny has a statistically equal probability of heads or tails?
I suggest that reasonable people will assume that the cherry-picked ten times was a normal statistical "blip" and that the coin is, in fact, "fair".
Apologists for pseudomedicine, which include the present Secretary of State for Health, eschew the same reasoning when it comes to clinical trials. They focus on the equivalent of the trial that gave seven tails and proclaim that "some trials show that homoeopathy [or antidepressants, or some other pseudomedicine] works better than placebo". Somehow they refuse to acknowledge that it is the totality of trials -- the thousand coin-tosses -- that must be taken into account, not just the few subsets that show a balance in favour of their pet species of pseudomedicine.
Similarly, those who subscribe to belief systems like astrology systematically ignore the many trials that demonstrate that, as a predictive or descriptive tool, it is no better than random chance and, instead, focus on those whose statistical-blip outcomes favoured the belief system.
So why do so many otherwise intelligent people seem reluctant to apply the same
reasoning to beliefs like astrology or the efficacy of pseudomedicine as they effortlessly do to coins?
The short answer is confirmation bias, probably reinforced by backfire effect. The unfortunate consequence is that those of us who are open to rational argument are unlikely to ever be able to convince those that are not.
Unfortunately, this also feeds into things like arguments for gun control, so it is unlikely that the recent appalling events in Connecticut will do anything other than polarise opinion even further.
And, following this argument to its rational conclusion, why did I even bother even to write this blog post?
I would be delighted to be shown to be wrong.