Applicatiion of Bayes' Theorem

Here is an extract from Daniel Kahneman's book "Thinking, Fast and Slow"

A cab was involved in a hit-and-run accident at night.  Two cab companies, the Green and the Blue operate in the city. You are given the following data:

• 85% of the cabs in the city are Green and 15% are Blue
• A witness identified the cab as blue. The court tested the reliability of the witness under the circumstances that existed on the night of the accident and concluded that the witness correctly identified each one the colors 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was blue rather than green?

Here is my solution:
        Let B denote the event that a Blue was involved
        Let G denote the event that a Green was involved
        Let A denote the event that a cab is identified as Blue.
We are asked to compute P(B/A).Using Bayes' Theorem we have

P(B/A) = (P(A/B)*P(B)) /  P(A)

Also,

P(A) = P(A/B)*P(B) + P(A/G) * P(G)

We know that

P(A/B) = 0.8
P(B)    =  0.15
P(G)   =   0.85
P(A/G) = 0.2     ...............(1.0 - 0.8)

Hence P(A) = 0.29
Hence P(B/A) = 0.15*0.8/0.29 = 0.414

The answer according to the book is 41%.