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Peter_de_Blanc comments on Open Thread: February 2010 - Less Wrong
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Bleg for assistance:
I’ve been intermittently discussing Bayes’ Theorem with the uninitiated for years, with uneven results. Typically, I’ll give the classic problem:
3,000 people in the US have Sudden Death Syndrome. I have a test that is 99% accurate; that is, it will wrong on any given person one percent of the time. Steve tests positive for SDS. What is the chance that he has it?
Afterwards, I explain the answer by comparing the false positives to the true positives. And, then I see the Bayes’ Theorem Look, which conveys to me this: "I know Mayne’s good with numbers, and I’m not, so I suppose he’s probably right. Still, this whole thing is some sort of impractical number magic." Then they nod politely and change the subject, and I save the use of Bayes’ Theorem as a means of solving disagreements for another day.
So this leads to my giving a very short presentation on the Prosecutor’s Fallacy next week. The basics of the fallacy are if you’ve got a one-in-3 million DNA match on a suspect, that doesn’t mean it’s three million-to-one that you’ve got that dude’s DNA. I need to present it to bright, interested people who will go straight to brain freeze if I display any equations at all. This isn’t frequentists-vs.-Bayesians; this is just a simple application of Bayes’ Theorem. (I suspect this will be easier to understand than the medical problem.)
I’ve read Bayesian explanations, but I’m aiming at people who are actively uninterested in learning math, and if I can get them to understand only the Prosecutor’s Fallacy, I’ll call Win. A larger understanding of the underlying structure would be a bigger win. Anyone done something like this before with success (or failure of either educational or entertainment value?)
If you're using Powerpoint, you might want to make a slide that says something like:
2,999 negatives -> 1% test positive -> 30 false positives
1 positive -> 99% test positive -> 1 true positive
So out of 31 positive tests, only 1 person has SDS.
If you've got the time, use a little horde of stick figures, entering into a testing machine and with test-positive results getting spit out.
Your numbers have me confused. I'd read the grandparent as implying 300M total population, out of which 3000 have the disease. (This is a hint to clarify the info in the grandparent comment btw - whether I've made a dire mistake or not.)
Another point to clarify is that the test's detection power isn't necessarily the inverse of its false positive rate. Here I assume "99%" characterizes both.
What I get: 300M times 1% false positive means 3M will test positive. Out of the 3000 who have the disease 30 will test negative, 2970 positive. Out of the total population the number who will test positive is 3M+2970 of whom 2970 in fact have the disease, yielding a conditional probability of .98 in 1000 that Steve has SDS.
I fail at reading. I thought it said "ONE in 3000 people in the US...."