You are a juror in a criminal trial, where the defendant is accused of robbing a

Question

You are a juror in a criminal trial, where the defendant is accused of robbing a bank. Testimony is offered that someone was seen fleeing the bank in a red car, and the DA points out that the defendant does in fact drive a red car. Assume the following: i. Assuming the defendant is in fact guilty, someone would have been seen in a red car with probability 1 (since he does own a red car, that's the only car he owns, he is a self-described loner, and based on your observation he seems dumb enough to potentially use his own car as the getaway car) ii. Assuming the defendant is in fact not guilty, there is a 14% probability that the person seen fleeing the bank would have been driving a red car (perhaps 14% of the cars registered in the area are red). iii. Prior to hearing this evidence, you believed there was a 25% probability the defendant was guilty based on other testimony. Use Bayes' Rule to find P(GIE), the probability the defendant is guilty given the evidence E that someone was seen leaving the bank in a red car. Same question if you believed the prior probability of the defendants guilt was only 1% (instead of 25%).

Explanation / Answer

By Bayes' Rule :

P(G/E) = P(E/G) P(G) / P(E)

P(E/G) = P(given that he is guilty, prob. that he was leaving in a red car) = 1 (Given)

P(G) = P(prob. that he is guilty) = 25% = 0.25 (Given)

-> P(E) = P(E/G)P(G) + P(E/G')P(G')

P(E/G') = P(given that he is not guilty, prob. that he was leaving in a red car) = 14% = 0.14 (Given)

P(G) = P(prob. that he is guilty) = 1 - 25% = 0.75 (Given)

P(G/E) = P(E/G) P(G) / P(E)

P(G/E) = (1)(0.25) / (1)(0.25) + (0,14)(0.75)

P(G/E) = 0.25 / 0.25 + 0.105 = 0.25/0.355 = 0.704

Ans : 0.704

If P(G) = 1% = 0.01 instead --->

P(G/E) = P(E/G) P(G) / P(E)

P(G/E) = (1)(0.01) / 1(0.01) + 0.14(0.99)

P(G/E) = 0.01 / 0.01 + 0.1386

P(G/E) = 0.01/0.1486 = 0.0673

Cheers!

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