The Cab Problem

[PrayingMantis]

I haven't seen this problem discussed here. If it was - I apologize. There seem to be some different opinions with regard to the solution.

A cab was involved in a hit and run accident. Two cab companies, the Green and the Blue, operate in the city. You know that:

(a) 85% of the cabs in the city are Green; 15% are Blue.
(b) a witness says the cab involved was Blue.
(c) when tested, the witness correctly identified the two colors 80% of the time.

The question is, How probable is it that the cab involved in the accident was Blue, as the witness reported, rather than Green?

[BruceZ]

[ QUOTE ]
I haven't seen this problem discussed here. If it was - I apologize. There seem to be some different opinions with regard to the solution.

A cab was involved in a hit and run accident. Two cab companies, the Green and the Blue, operate in the city. You know that:

(a) 85% of the cabs in the city are Green; 15% are Blue.
(b) a witness says the cab involved was Blue.
(c) when tested, the witness correctly identified the two colors 80% of the time.

The question is, How probable is it that the cab involved in the accident was Blue, as the witness reported, rather than Green?

[/ QUOTE ]

Standard Bayes' theorem problem. See this more complicated one asked the other day.

P(blue | witness says blue) = P(blue AND witness says blue) / [P(blue AND witness says blue) + P(green AND witness says blue)]

= 0.15*0.8 / (0.15*0.8 + 0.85*0.2)

= 12/29.

[PrayingMantis]

Bruce,

The Bayesian reasoning behind this solution is clear, however, I think there is at least some controversy about it, that's why I posted it here, to see if an interesting discussion could be developed. Personally, I'm interested in reading different views about this.

This is a quote from this article ("Cohen" is L.J Cohen):
http://www.ed.uiuc.edu/eps/pes-yearb...s/MCCARTHY.HTM

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Cohen, however, rejects the probabilistic reasoning of Kahneman and Tversky. He argues that the .41 probability figure reached by Kahneman and Tversky is merely “…the value of the conditional probability that a cab-color identification by the witness is…[correct], on the condition that it is an identification as…[blue].” And this is not what we need to know.


Rather, the issue is “…the probability that the cab actually involved in the accident was blue, on the condition that the witness said it was [blue].” According to Cohen, “if the jurors know that only 20% of the witness’s statements about cab colors are false, they rightly estimate the probability at issue as [80%]…the fact that cab colors vary according to an 85/15 ratio is strictly irrelevant….”


The jurors, says Cohen, should be interested only in the “causal propensity” of the witness to correctly identify cab-colors, and this is dependent only on “causal properties, such as the physiology of vision, [which] cannot be altered by facts…that have no causal efficacy;…the mere relative frequency of blue and green cabs…does not generate any causal propensity for the particular cab in the accident.” Hence, by adopting a “propensity account” of probability, Cohen vindicates the ordinary “common-sense” judgments of the man in the street: not surprisingly, his conclusion has a great deal of intuitive plausibility.


The propensity interpretation of probability used by Cohen seems particularly plausible in this scenario, since under this interpretation probabilities can properly be attributed to particular individuals. The difficulty is that, given this account, there is no clear way to determine just how the various operative factors contribute to the final propensity. According to Cohen, “the main weakness of a propensity analysis is that it does not intrinsically carry with it any distinctive type of guidance in regard to the actual evaluation of probabilities…since the talk of propensities has no distinctive numerical implications, it provides no inherent basis for the assignment of actual probability-values.”

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[alThor]

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Personally, I'm interested in reading different views about this.

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Right one and wrong ones? [img]/images/graemlins/smile.gif[/img]

Use Cohen's reasoning (or rather, the reasoning the author attributes to Cohen) in the following version: (change are in boldface)

A cab was involved in a hit and run accident. Two cab companies, the Green and the Blue, operate in the city. But the day of the accident, Blue Cab drivers were all on strike. You know that:

(a) 100% of the cabs in the city are Green that day; 0% are Blue.
(b) a witness says the cab involved was Blue.
(c) when tested on other days, the witness correctly identified the two colors 80% of the time.

The question is, How probable is it that the cab involved in the accident was Blue, as the witness reported, rather than Green?

Cohen (according to the author of your article) still says 80%. What do you think?

(And I don't ask mathematicians philosophy questions either... [img]/images/graemlins/smile.gif[/img] )

[PrayingMantis]

alThor,

First, I would say this: in your variation of the problem - it seems unreasonable to even ask a witness, since if you accept (a), it is clear that the cab involved was green. However, in the original problem, the witness has an important role (or according to Cohen according to the article - the only role) in deciding which was the color of the cab.

I think that the argument against using bayesian reasoning in the original problem does make some sense. I'll give you another (my own) version of the problem ( (a) and (b) are the same as in the original problem):

(c) when answering questions, the witness lies 20% of the time and tells the truth 80% of the time.

The question is, How probable is it that the cab involved in the accident was Blue, as the witness reported, rather than Green?


Also, as to your comment about mathematicians and philosophers, I think that there are many aspects in probability that ask for a more, lets say, epistemological attitude than most other fields in mathematics. However, I'm definitely not a mathematician or a philosopher.

[PairTheBoard]

I think alThor's point is spot on. Suppose the prosecutor brings the witness who says he saw a blue cab and an expert who testifies that under controlled tests the witness has the 80% accuracy.

At that point the jurors will weigh the evidence according to the 80% accuracy. But suppose the Defense then brings an expert who testifies that 0% of the cabs running that day were Blue. Clearly the Jurors will take that evidence into account and conclude the witness was mistaken. But now suppose instead that the Defense brings an expert who testifies that only 1% of the cabs running that day were Blue. Should the jurors not also take that evidence into consideration to conclude there is a high likelihood the Witness is mistaken? Surely even Cohen would agree they should. But what if it's 2%. Or 3%. Or 15%? At what point does Cohen argue that such evidence becomes irrelevant?

PairTheBoard

[PrayingMantis]

[ QUOTE ]
I think alThor's point is spot on. Suppose the prosecutor brings the witness who says he saw a blue cab and an expert who testifies that under controlled tests the witness has the 80% accuracy.

At that point the jurors will weigh the evidence according to the 80% accuracy. But suppose the Defense then brings an expert who testifies that 0% of the cabs running that day were Blue. Clearly the Jurors will take that evidence into account and conclude the witness was mistaken.

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In this point of the trial it simply seems that the witness and the expert testifying about his accuracy are both completely irrelevant. The cab was green since it couldn't have been blue.

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But now suppose instead that the Defense brings an expert who testifies that only 1% of the cabs running that day were Blue. Should the jurors not also take that evidence into consideration to conclude there is a high likelihood the Witness is mistaken? Surely even Cohen would agree they should. But what if it's 2%. Or 3%. Or 15%? At what point does Cohen argue that such evidence becomes irrelevant?

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This is an interesting perspective. I might think that in a way, as strange as it might sound, the more blue cabs running at the time of the accident (up to some point), the more relevant is the witness. Cohen is probably exaggerating when arguing that the witness is all that matters, but maybe there's some way of assessing how relevant is the witness, as some function of the base rate?

[jay_shark]

What if some of the cabs are colored green and blue :P

The cabs must be strictly green or blue but not both . This is important if you want to use Bayes theorem for the calculations.

[PrayingMantis]

Also, another comment with regard to alThor's variation:

When we know that there were 0% blue cabs running at the time of the accident, we can conclude that the witness is now wrong, that is, we know that we are in the X% of the time where he confused between the colors, so it doesn't matter how often he is right or wrong about it, i.e, this information becomes technically unnecessary for solving the problem. He might as well be right 99% of the time and the answer still does not change.

[pzhon]

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According to Cohen, “if the jurors know that only 20% of the witness’s statements about cab colors are false, they rightly estimate the probability at issue as [80%]…the fact that cab colors vary according to an 85/15 ratio is strictly irrelevant….”

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This is hideously wrong. It should not be a matter of debate.

Interestingly, the same mistake is also behind the envelope paradox, where it is less obviously a mistake. I was thinking of using the cab problem or the disease test result to try to explain the fallacy. That there is a 50% chance that you receive either envelope doesn't mean the conditional probability that what you see is the higher amount is 50%, given the amount. That's not very intuitive, but it is more intuitive that if a 99% accurate test says that you have a rare disease, you don't think you have a 99% chance of having the disease. You suspect a false positive, and the probability that you actually have the disease depends on its rarity even if you don't know exactly how rare it is.