Take the big question: is Christianity overall good for us or overall bad for us? A Christian will look around and say that 95% of the Christians are good, therefore it's overall good. An atheist will look around and see that 95% of the bad people are Christians, so it's overall bad. Obviously, both claims can be true at the same time, so how do we make sense of it? Well, it doesn't really matter so much if 95% of Christians are good when only 1% of all people are bad. If Christians hardly ever encounter bad people in their daily lives, they think it's because Christianity is working and so it must be a good test of goodness. And they're just plain wrong. It gets really bad when one of these 1%-ers comes along, like Bush or Pat Robertson, and they all give these bad ones unconditional support because they passed the Christianity litmus test. Guys like Bush, even guys like Hitler, have all been helped immensely because people suck at applying Bayes' rule to their daily lives. Instead, people actually need to be taught it and even then it's hard to understand the implications.

How reliable would the test have to be before a positive result would give us 95% confidence that the testee was infected with the 1-in-100 disease?

99.947%

Ebonmuse, I think there's a typo in your original post:
"Adding up these terms, we get an overall P(A) of...." - shouldn't this be "...overall P(B) of..."?

Of course, having a one-in-five chance of dying sometime in the next year might be, to some people, enough reason to draft a will anyway.

So, follow-up question. How reliable would the test have to be before a positive result would give us 95% confidence that the testee was infected with the 1-in-100 disease?

There was a case in Britain, I believe, a few years back where a woman was convicted of murder after three of her children suffered SIDS, though there was no specific evidence of homicide.

Just for completeness the person in question was sally Clark , a lawer who was convicted of killing two of her children who apparently died from SIDS. The expert witness was subsequently pilloried for mis representing the stats, sadly Sally Clark, although being aquited in 2003 has subsequently died.

Actually the fallacy here was the assumption that the events were statistically independent.

Actually, that's just one of the errors here; there may also be other risk factors as well. Ebonmuse pointed out the most egregious error, though: long odds simply do constitute evidence. Any event having a non-zero probability is almost certain to occur given enough opportunities. In the absence of positive evidence of parental misconduct, the a priori probability of three infants in the same family dying of SIDS is simply irrelevant if that's what actually happened.

Now I feel better. :)

Talk about relativity...

There was a case in Britain, I believe, a few years back where a woman was convicted of murder after three of her children suffered SIDS, though there was no specific evidence of homicide. The prosecutor argued that it was far too unlikely that three infant deaths in a row in the same family could be anything other than malice. But neither he nor the jury ever considered the question of how many families we'd expect that to happen to just by chance.

Actually the fallacy here was the assumption that the events were statistically independent. Say the probability of SIDS is 1/1000, three deaths would be 1/1000 x 1/1000 x 1/1000 or 1/1000000000. In fact if you allow that there could be genetic abnormalities that contribute to SIDS it is actually quite likely that more than one event could happen in a familly. i.e the events are not statistically independent.

So if there are statistical problems in DNA testing and matching and it's not always conclusive....then what's Maury Povich gonna do?????

Maury hasn't got anything to worry about, actually. Different statistics apply to paternity tests. Everybody gets exactly half of their genome each from their mother and their father, so if you have a sample of a person's DNA, then half of the genetic markers will be identical to something in the mother's genome, and half will be identical to dad's (ignoring mutations). Since the identity of the mother is generally not in question, for obvious reasons, you can just cancel out the mother's genetic markers and see whether dad has something to match the rest. It's possible that two people fit the father markers, but if you have a small sample (say, all the men that the mother had sex with nine-ish months before the kid was born), it's not likely to come up.