The Cab Problem

[EverettKings]

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Here's a more intuitive explanation for anyone still unclear.


The heart of the matter is that witnesses are simply exposed to a green cab more often than a blue one. They are given more chances to falsely report a green cab as blue than to report a blue cab as blue.

To reason that "witnesses make correct observations 80% of the time, therefore the cab was blue 80% of the time" ignores information, specifically the distribution of green and blue cabs.

The witness is simply more likely to be presented a green cab and call it blue than to see a blue cab in the first place. Simple as that.



Everett

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For someone trying to summarize, you did an excelent job of completely missing the point.

Bravo [img]/images/graemlins/laugh.gif[/img]

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My understanding after reading this thread is that you are one of the few that is still missing the point.

Can you come up with a simple explanation of your point? I thought my ~100 words was a perfect explanation of what's going on in the math, and I'd love if you could directly prove my arguments wrong.

[pzhon]

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Can you come up with a simple explanation of your point? I thought my ~100 words was a perfect explanation of what's going on in the math, and I'd love if you could directly prove my arguments wrong.

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Your math looked fine, given one model. Your model is explicitly spelled out in some formulations of the cab problem.

It looks like SplawnDarts is raising A point, not THE point. He is saying that the way the OP phrased the problem is ambiguous, and that this can be an explanation for some alternate answers. Since there is about as much confusion even when this ambiguity is removed, I don't think this is the primary issue, but perhaps SplawnDarts is saying that even when the conditional information is expressed clearly, people misread it and assume something else.

There are other ambiguities, which are less frequently spelled out. (This is typical in word problems. Part of mathematical modelling is identifying these added assumptions.) It is necessary to assume something about the likelihood that each cab driver is involved in the accident. I live close to a cab company, and they are disproportionally represented near my location. Other cab companies may tend to have more experienced or better trained drivers. This ambiguity isn't behind the cab problem. The difficulty comes from our inconsistent and faulty methods for handling information about probabilities.

[pzhon]

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The evidence that the cab was green was stronger. If the witness had tested to be more reliable, this would not have been the case.

Many criminal cases (where the standard of proof is much higher) are based on personal identifications across races, even though people usually have difficulty identifying people of other races. We have a bias toward giving too much weight to eye-witness testimony in circumstances where it is not warranted. That leads to horrendous injustices.

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Here is a relevant ruling:

"The court ruled that because prosecutors had convicted Clarence based on that compelling eyewitness testimony, a jury would’ve reached the same conclusion even if it had known his DNA didn’t match the killer’s. "

This sounds like an example of combining an overestimation of the reliability of eye-witness testimony (in this case, by a traumatized 6-year-old) with a failure to understand Bayesian analysis.