A witness testifying in a hit-and-run case indicates that a green cab hit the pl

Question

A witness testifying in a hit-and-run case indicates that a green cab hit the plaintiff in the case. You are aware that 85% of the cabs in the city are blue and the other 15% are green. However, the defense attorney conducts an experiment recreating the conditions at the scene, and the witness (when shown green and blue cabs) is only able to correctly differentiate them 80% of the time. Thus, what is the probability that in fact a green cab (to say nothing about the green cab driven by the defendant!) was involved in the accident? Assume that prior to the witness' testimony, you would believe there was a 15% probability that a green cab was involved. (source : Thinking Fast and Slow by D. Kahneman, p. 161)

Explanation / Answer

Let G be the event when the culprit is Green Cab and B be the event when the Culprit is Blue Cab.

Let W be the witness report.

Given, p(G) = 0.15, p(B) = 0.85

Using Bayes Theorm,

P(G|W) / P(B|W) = {P(W|G) / P(W|B)} * {p(G)/p(B)}

= {0.8/0.2} * {0.15/0.85}

= 12/17

Hence,

P(G|W) = (12/17) * P(B|W)

Also,

P(G|W) + P(B|W) = 1

Hence,

P(B/W) = 0.586207

and P(G/W) = 0.413793

Ans: 0.413793 or 41.38%

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