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?

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