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A blue taxi (Posted on 2002-05-24) Difficulty: 4 of 5
A crime has occured in Carborough, involving a taxi. The police interviewed an eyewitness, who stated that the taxi involved was blue.

The police know that 85% of taxis in Carborough are blue, the other 15% being green. They also know that statistically witnesses in these situations tend to be correct 80% of the time - which means they report things wrong the other 20% of the time.

What is the probability that the taxi involved in the crime was actually blue?

See The Solution Submitted by levik    
Rating: 3.7857 (14 votes)

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A more paradoxical version. | Comment 10 of 17 |
The more paradoxical version of this puzzle results if the witness stated that the taxi involved was green. This is the way that works out:

As before, let's say the city has a total of 100 taxis (this way we can have a one to one relationship between taxis and percent).
Of these 100, 85 are blue, and 15 are green.

Again, let's look at both cases:

85/100 blue taxi is involved. Since the witness will be wrong 20% of the time, they will say they saw a blue taxi 68 times, and claim that the taxi was green the other 17 times.

15/100 green taxi is involved. 12 witnesses would correctly identify a green taxi, but 3 would wrongly claim to have seen a blue one.

This time we know that the witness said they saw a green taxi. There is a total of 17+12 = 39 percent chance for that to happen. Of those 39%, 12% of the time the taxi will in truth be green. Thus the probability of the taxi being green is 12/39 = approximately 30.8%.

Thus it is only about 31% likely that the color of the cab agrees with the witness's description.

  Posted by Charlie on 2003-03-26 14:09:57
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