Based on the evidence given, we can calculate a probability of 41% that the cab was blue. This witness is pretty reliable, but blue cabs are rare and it is more likely that the witness is mistaken than that the cab was really blue. My approach was to imagine 100 hit-and-run accidents. In 85 of them, the cab will be green, in 15 blue. Of the 85 green accidents, the witness will correctly see 67 green cabs (eighty percent accuracy) and mistakenly see 17 blue cabs. Of the 15 blue accidents, the witness will correctly see 12 blue cabs and mistakenly see 3 blue cabs. We are told that the witness saw blue. So we are considering one of the 17 mistaken identifications or 12 correct identifications, a total of 29 cases. The witness is correct (and the cab is really blue) in 12 out of 29 cases, about 41%.
I'm glad we arrived at he same answer. I suppose lil will have a bit easier time following your logic. I had only read the bullet points from your post, and not the text of the book page, so I think I may have rehashed some unnecessary stuff. Anyway, lil, the main point is that there are a lot of counter intuitive things in statistics.
Yes, and our number agrees with the footnote in the book, but I still don't follow the explanation there. Where did the number 1.706 come from? Simpson's Paradox is my favorite statistical anomaly.
I'm not sure either. I had tried to manipulate the equation to read something of the form P(A)/(1 + P(A)), but I couldn't find an obvious way to do that. Also, it doesn't work the other direction. That is, the probability of the car being green when it's reported green does not equal the probability of the car being reported blue divided by 1 plus that probability, so it's certainly not a general solution to the problem, but perhaps a weird coincidence of the way he did the arithmetic.Where did the number 1.706 come from?