The avclub link makes more errors than I can count, honestly. For one, you don't combine the probabilities unless they're independent. So for example, the kid losing his knife and the murder being committed by the same kind of knife aren't independent, because it's plausible that the kid left the knife at his father's place and the actual killer used it to commit the murder.
The next problem is that prosecutors systematically reinforce confirmation bias in presenting their case. The probability that someone has brown eyes and is 5'7" tall and lives in New York and knows the victim and has a motive etc. etc. would seem to be strong evidence (especially if you're assuming them all to be independent, which they're not), but that only works if you're taking an unbiased sample of the possible characteristics of the perpetrator, which the prosecutor explicitly does the opposite of. If there is no apparent motive, or the knife used to kill the victim isn't a match for the one the defendant is known to have, or the perpetrator was wearing a suit and the defendant isn't known to own a suit, you won't hear any of that from the prosecution. And if you only consider the things that match, using the article's flawed method, all additional evidence can only ever increase the probability of guilt, since any evidence to the contrary doesn't make it into the calculation. All you have to do is keep collecting evidence and excluding anything that doesn't comport with the prosecution's theory of the crime and soon you have a seemingly insurmountable case.
Then you have the "DNA database" problem with statistics. Take a 1 in 10,000 chance and it sounds like solid proof ("99.99%") but if the population you're testing against is 6,000,000,000 people then you still have a pool of suspects containing 600,000 people. You can't then just pick one of those people arbitrarily and claim a 99.99% chance that that was the perpetrator, the probability that a person is the actual perpetrator if chosen at random from the group of individuals whose DNA would match is only ~0.000167%. It isn't good enough to prove that a defendant is statistically in the top thousand people in the city as far as probability to have been the perpetrator of a particular crime. You still have to exclude all the others or you'll convict the wrong man.
The next problem is that prosecutors systematically reinforce confirmation bias in presenting their case. The probability that someone has brown eyes and is 5'7" tall and lives in New York and knows the victim and has a motive etc. etc. would seem to be strong evidence (especially if you're assuming them all to be independent, which they're not), but that only works if you're taking an unbiased sample of the possible characteristics of the perpetrator, which the prosecutor explicitly does the opposite of. If there is no apparent motive, or the knife used to kill the victim isn't a match for the one the defendant is known to have, or the perpetrator was wearing a suit and the defendant isn't known to own a suit, you won't hear any of that from the prosecution. And if you only consider the things that match, using the article's flawed method, all additional evidence can only ever increase the probability of guilt, since any evidence to the contrary doesn't make it into the calculation. All you have to do is keep collecting evidence and excluding anything that doesn't comport with the prosecution's theory of the crime and soon you have a seemingly insurmountable case.
Then you have the "DNA database" problem with statistics. Take a 1 in 10,000 chance and it sounds like solid proof ("99.99%") but if the population you're testing against is 6,000,000,000 people then you still have a pool of suspects containing 600,000 people. You can't then just pick one of those people arbitrarily and claim a 99.99% chance that that was the perpetrator, the probability that a person is the actual perpetrator if chosen at random from the group of individuals whose DNA would match is only ~0.000167%. It isn't good enough to prove that a defendant is statistically in the top thousand people in the city as far as probability to have been the perpetrator of a particular crime. You still have to exclude all the others or you'll convict the wrong man.