cab puzzle (from FAQ)

I came across this in the FAQ when I was looking for an explanation of derangements and found the hat check puzzle. I thought some here might never have seen it.

A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. Here is some data: a) Although the two companies are equal in size, 85% of cab accidents in the city involve Green cabs and 15% involve Blue cabs. b) A witness identified the cab in this particular accident as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colours 80% of the time and failed 20% of the time. What is the probability that the cab involved in the accident was Blue rather than Green? If it looks like an obvious problem in statistics, then consider the following argument: The probability that the colour of the cab was Blue is 80%! After all, the witness is correct 80% of the time, and this time he said it was Blue! What else need be considered? Nothing, right?

Or wrong?

cheers

DDEckerslyke schrieb:

Ok, so it's 80% for Blue, but that doesn't take into account that 20% might be wrong, so it's actually 60% for Blue, and as we know from the problem, 85% for Green, which means the chances are 60:85 in favor that it actually was a Green cab!

Whaddaya mean, 'It's not a proper explanation'?

Ooops.

That was why I posted both answers

Imagine a vampire-detecting device. When focused on a vampire, it correctly identifies them as a vampire 99.9% of the time and mistakenly identifies them as a human 0.1% of the time. When focused on a human, if correctly identifies them as a human 99.9% of the time and mistakenly identifies them as a vampire 0.1% of the time. The device is set up at an entrance to Wrigley Field before a home game, and after about a thousand people go in it identifies one as a vampire. What is the probability that the device is correct? Your answer would seem to be 99.9%, since that is the accuracy level of the device.

There was a time when that would have been a trick question, as no vampires would ever attend a game at Wrigley Field. But alas, the CUBS (Citizens United for Baseball in the Sunshine) failed in their efforts to keep lights (and vampires) out of Wrigley

True, but I think people still know what I meant by that.

Can we even answer that question if we don't know the ratio of vampires / humans at Cubs games?

Spoiler below.

Eamon Warnock schrieb:

'Well, how certain are we that the Red gang did it? For if the Grenn gang didn't, the Red surely must have! If the chance that a Red is to blame is less than 5%, then the chance against Green must be more than 95%. 'There's a chance of 20% that the murderous gang member was a Red; there's 15% chance that the witness identified him as Green; 20% * 15% = 3% overall. This fails to reach the 5% limit, hence we can safely assume the murdere was not a Red gang member: he must've been a Greenie!'

The defense lawyer appears thunderstruck. However, when the commotion has died down, he composes himself, rises, and proceeds to smash the prosecution's case. How?

I can guess, but the defense should have pointed out that while we know there's 4 times as many Greens as Reds, we have no information about the likelyhood of a Red committing a crime compared to that of a Green, and thus the requisite certainty could not be met.

Well, whereas the Reds are merely 'notorious', the Greens are known to be 'ruthless'.

Anyway, 1) Green committed murder, witness was right. 0.8 x 0.85 = 0.68

2) Green committed murder, witness was wrong. 0.8 x 0.15 = 0.12

3) Red committed murder, witness was right. 0.2 x 0.85 = 0.17

4) Red committed murder, witness was wrong. 0.2 x 0.15 = 0.03

Given that witness has identified Green, only outcomes 1 and 4 are possible.

Probability of Green being guilty is then 0.68/(0.68+0.03) = 0.958

But unless I screwed up, that still puts him above the 95% threshhold. I say give him the chair anyway. As the old saying goes, it woud be an injustice to execute him for the wrong crime, but a far graver one not to execute him at all.