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September 12th, 2009, 02:19 PM   #1
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simple probability question

Hi,

how do I solve the following question:

Exactly 2 cab companies opearate in Belleville. The Blue Company has blue cabs and the green Company has Green Cabs. Exactly 85% of the cabs are blue and the other 15% are green. A cab was involved in a hit-and-run accident at night. A witness, Wilbur, indetified the cab as a Green cab. Careful tests were done to ascertain people's ability to distinguish between blue and green cabs at night. THe tests showed that people were able to indentify the color correctly 80% of the time, but they were wrong 20% of the time.
What's the probaiblity that the cab involved in the accident was indeed a green cab ?

thanks
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September 13th, 2009, 03:22 PM   #2
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There are many minor assumptions that are needed in order to find the answer. For example, you need to assume that such cab identifications are always as blue or green, never as red, say. I won't list them all!

I'll assume that the prior probability that the cab is green is 15% (which wouldn't be the case if, for example, no green cabs are used at night, or if all drivers of green cabs are sufficiently law-abiding never to be involved in hit-and-run accidents).

You are supposed to calculate that the prior probability that a cab in the circumstances described (except for the specific color) is identified (correctly or incorrectly) as green is 80% of 15% + 20% of 85%, i.e., 29%. Although it's easy to see how I did that calculation, it's a bit tedious to precisely specify the minor assumptions needed to justify it, so I'll simply assume that 29% is correct.

You are supposed to calculate that the probability that a green cab in the circumstances described (except for the specific color) is identified as green is 80%. Again, it's a bit tedious to specify the minor assumptions needed for that, so I'll assume that 80% is correct. It could be incorrect if witnesses tend to identify color better during daytime than at night.

We've now got enough data to apply Bayes's theorem and hence calculate the answer as 80% of 15% divided by 29%, i.e., 12/29. (Or (1200/29)% if you prefer percentages to fractions.)
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September 13th, 2009, 10:36 PM   #3
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Re: simple probability question

ok thanks a lot
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