GATE EC 2018 Set 1 — Question 9
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Engineering Mathematics → Probability → Bayes' Theorem
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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?
(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?
Correct answer
(C) 41%
Solution
This is a problem of conditional probability that can be solved using Bayes' theorem.
Let's define the events:
The closest option is 41%.
Let's define the events:
- : The event that the cab involved in the accident is Blue.
- : The event that the cab involved in the accident is Green.
- : The event that the witness identifies the cab as Blue.
1.Prior probabilities of the cab colors:
- (85% of cabs are green)
- (15% of cabs are blue)
- The witness is correct 80% of the time. This means:
- (Probability the witness says Blue, given the cab was Blue)
- (Probability the witness says Green, given the cab was Green)
- The witness is incorrect 20% of the time. This means:
- (Probability the witness says Blue, given the cab was Green)
The closest option is 41%.
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