# Use Bayes' methods to find the probability that the cab involved in the accident was actually Blue

A cab driver was involved in a deadly hit-and-run accident at night. Two cab companies, the Green and the Blue, operate the city; 15% of the cabs are Green and 85% are Blue. A witness identies the cab as Blue. The court tests the reliability of the witness under the same circumstances that existed on the night of the accident and concludes that the witness can correctly identify the color of the cab 80% of the time.

-

We know that $P(G) = 0.15$ and $P(B) = 0.85$ where $G$ and $B$ denote the proportion of green and blue cabs respectively. Also $P(IB|B) = P(IG|G) = 0.8$ where $IB$ and $IG$ denote "identifying blue" and "identifying green" respectively.
So $$P(B|IB) = \frac{P(B \cap IB)}{P(IB)}$$
$$= \frac{(0.8)(0.85)}{(0.8)(0.85)+(0.2)(0.15)}$$