In the city of M, there are only taxis in two colours, red and blue. One night a cab was involved in a hit-and-run incident. As per a witness, the colour of the cab was blue. Based on the information from the authorities, 80% of the cars are red, and 20% are blue. Tests have found that the accuracy at which the witness can identify the colours is about 80% under challenging lighting conditions. What is the probability that the witness correctly identified the right one?
Well, we will use Bayes’ rule to estimate the accuracy. Here is Bayes’ rule modified to suit our context.
The terms are
P(B|W) - the probability that the cab colour is blue, given the witness' testimony.
P(W|B) - the probability that the witness identifies blue, given the cab is blue = 80% or 0.8.
P(B) - a priori probability of finding a blue cab in the city = 20% or 0.2.
P(W|R) - the probability that the witness identifies blue, given the cab is red = 20% or 0.2.
P(R) - a priori probability of finding a red cab in the city = 80% or 0.8.After substituting the numbers in the equation, the required probability becomes:
No different from tossing a coin!
