A cab was involved in a hit and run accident at night

Q. A cab was involved in a hit and run accident at night. You are given the following data about the cabs in the city and the accident.

i. 85% of cabs in the city are green and the remaining cabs are blue.

ii. A witness identified the cab involved in the accident as blue.

iii. It is known that a witness can correctly identify the cab colour only 80% of the time.

Which of the following options is closest to the probability that the accident was caused by a blue cab?

(A) 12%                          (B) 15%                      (C) 41%                      (D) 80%

Ans: 41%

Sol:

Probability that cab is green P(G) = 0.85
Probability that cab is blue P(B) = 0.15
Probability that witness is correct = witness identifies blue cab = P(W/B) = 0.8
Probability that witness is wrong = witness identifies blue cab = P(W/G) = 0.2
Probability that accident was caused by blue cab provided witness identifies it

Total Probability = P(W) = P(B)P(W/B) + P(G)P(W/G)

= 0.15 × 0.8 + 0.85 × 0.2 = 1.2 + 1.7 = 2.9

P(B/W) = (0.15 × 0.8)/2.9
= 1.2/2.9 = 0.41 = 41%