There I was, putzing around the Internet, all beveraged up thanks to the wonderful town of Lynchburg, Tennessee, (pop. 361, if the label is to be believed), when I stumbled upon this wonderful problem I could not solve (did I mention the "beveraged up" part?):
There is a city which hosts two taxi-cab companies, the Blue Cab Co. and the Green Cab Co. Blue cabs are blue and Green cabs are green; they are otherwise identical. 70 percent of the cabs in the city are Blue cabs, and 30 percent of the cabs in the city are Green cabs. Moreover, historically speaking, Blue cabs have been involved in 70% of all traffic accidents in the city that involved cabs, and Green cabs have been involved in 30% of all traffic accidents in the city that involved cabs. One night, there is a traffic accident involving a taxi-cab in the city, to which there is one witness. Authorities perform extensive tests on the witness, and determine that his ability to recognize cabs by their color at night is approximately 80 percent accurate and 20 percent inaccurate (meaning that when he is wrong he does not say he doesn’t know, but rather misidentifies it as being of the other color). The witness says the taxi-cab involved in the accident was ‘blue.’ On these facts, and strictly assuming the taxi-cab was not from some other city, what is the approximate probability that the taxi-cab involved in the accident belonged to the Blue Cab Co.?
Enjoy your Statistics class, Amy. :D
Solution tomorrow. Ish.
Update: Solution is available in Comments
Posts
This strikes me as similar to the coin toss problem, i.e. if you toss a coin one hundred times and it comes up heads one hundred times what is the probability that the coin will come up tails on the next toss? The answer is, of course, 50%.
ReplyDeleteTherefore, the probability that a blue cab was involved in this individual accident is 70%.
I do not, however, understand how the accuracy of the witness changes the probability. I'm looking forward to the solution.
Your the math guy so why are you asking us? 56%
ReplyDeleteYou're
ReplyDeleteHere is the solution as given:
ReplyDeleteIf there were 100 accidents, we could assume from the base rate that 70 would involve Blue Cabs and 30 would involve Green Cabs. Assuming that a witness just like the one here (i.e., 80 percent accurate and 20 percent inaccurate) witnessed each accident, the 100 accidents would be reported as follows: of the 70 Blue-Cab accidents, the witness would say ‘blue’ (correctly) 56 times and ‘green’ (incorrectly) 14 times; of the 30 Green-Cab accidents, the witness would say ‘green’ (correctly) 24 times and ‘blue’ (incorrectly) 6 times. We are concerned here with the times the witness would say ‘blue.’ Of those 62 times, the witness would be correct 56 times. 56 over 62 = .903, or approximately 90 percent probability.
If it makes you feel better, my answer was 80%.
I found this posted in the comments at TierneyLab (comment #86, from someone calling himself “xaosdog”). There’s a bunch of fun puzzles amongst the comments. Plus a couple in the main entry.
I don't want to understand this....
ReplyDeleteIt is a known fact that cabs driven by people who look just like Al Sharpton have the least amount of accidents. Because no one will get in. ~Mary
ReplyDeleteMy brain hurts. I'm going to have to get drunk.
ReplyDeleteFrankandMary: In that case, I'll wait for a cab driven by someone who looks like Al Sharpton. Because it's safer.
ReplyDeleteAnna: That's what I do. :)