GATE 2018 EC GA Q.9: solution and answer

GATE 2018 · ECGA Q.9 · 2 marks · MCQ

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%

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Answer and solution

ANSWER

The probability that the cab was blue, given that the witness said blue, is $\frac{0.12}{0.29} \approx 41%$ , so the answer is (C).

Official answer: CSource: Official answer key, GATE 2018 EC (IIT Guwahati)

Prior probabilities (before the witness speaks): $P (G) = 0.85$ and, since every other cab is blue, $P (B) = 1 - 0.85 = 0.15$ .
The witness names the colour correctly 80% of the time. There are only two colours, so a wrong answer means naming the other one: $P (says blue ∣ B) = 0.8$ and $P (says blue ∣ G) = 1 - 0.8 = 0.2$ .
Total probability theorem: $P (says blue) = P (B) P (says blue ∣ B) + P (G) P (says blue ∣ G) = 0.15 \times 0.8 + 0.85 \times 0.2 = 0.12 + 0.17 = 0.29$
Bayes' theorem: $P (B ∣ says blue) = \frac{P ( B ) P ( says blue ∣ B )}{P ( says blue )} = \frac{0.12}{0.29} \approx 0.414$
So the probability is about 41%, which is option (C).

KEY IDEA

This is Bayes' theorem. A witness saying "blue" can come from two routes: a blue cab named correctly ( $0.15 \times 0.8 = 0.12$ ) or a green cab named wrongly ( $0.85 \times 0.2 = 0.17$ ). The answer is the share of the first route in the total, $\frac{0.12}{0.12 + 0.17}$ . Green cabs are so common that their 20% misidentifications outnumber the correct "blue" calls.

WATCH OUT FOR

Picking 80% (D) because the witness is 80% reliable, which ignores how few cabs are blue. Picking 15% (B) is the opposite mistake: it uses only the proportion of blue cabs and ignores the witness. Picking 12% (A) means stopping at $P (B) P (says blue ∣ B)$ without dividing by $P (says blue) = 0.29$ .

From the GATE 2018 EC paper. Every question of this paper · Download the full paper · All 7 Statistics & probability questions, with notes

GATE 2018 EC GA Q.9: solution and answer

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