Talk:Chance News 63
Re "Demystifying conditional probability"
Here is a comment, in response to Strogatz' "Chances Are" article in NYT, by blogger #17, Phil Price, Berkeley, CA:
This article is well written, and I like a lot of things about it...but I'm both perplexed and frustrated by your characterization of the "common sense" cancer calculation, because (except for the rounding) you have indeed applied Bayes' Theorem, whether you know it or not. Bayes' Theorem, applied to this example, says that to find the probability of cancer, given the positive mammogram, one should multiply the probability of having cancer by the probability of having a positive mammogram if cancer is present, and then divide by the probability of having a positive mammogram whether cancer is present or not. In mathematical notation, with "c" meaning "yes to cancer" and "m" meaning "positive mammogram", we have p(c | m) = p(c) p(m|c) / p(m)
This is the calculation you have done. You first multiplied the cancer rate (8/1000) times the probability of getting a positive mammogram if you have cancer (9/10) to calculate the numerator (about 7/1000). Then you did a little math to determine the total rate of positive mammograms out of every 1000 women whether they have cancer or not (about 77/1000). Then you divided, to get 7/77, or --- finally switching to the rate per 100 rather than the rate per 1000 --- about 9%.
Your calculation is EXACTLY as "labyrinthine" as Bayes' Theorem, because it is EXACTLY THE SAME as Bayes' theorem. Rather than telling your students that you have a better way of doing the calculation that avoids the complexities of Bayes' Theorem --- a claim that isn't true --- you should tell them that you can explain why Bayes' Theorem makes sense. You've done a good job at that.
Submitted by Margaret Cibes
- This is quite right. BUT, the calculation that Strogatz gives (basically Gigerenzer's "natural frequencies" approach) is a lot easier to explain to mathematically naive people. I speak from many years' experience doing this. In particular, it is easy to explain simple problems like the mammogram example without using pencil and paper to people like this. I've done it many times. Sure, it is mathematically equivalent to Bayes' theorem, but you should see the lights go on when I explain these examples to people in the parking lot, on the plane, etc. Bill Jefferys 18:53, 7 May 2010 (EDT)