Question

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%

Answer 1

The correct option is C 41%

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

$P\left(B/W\right)=\frac{P(B\cap W)}{P\left(W\right)}=\frac{P\left(B\right)P\left(W/B\right)}{P\left(W\right)}\text{Baye's theorem}$

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%