In a jury trial, suppose that in order for the defendant to be convicted, it tak

Question

In a jury trial, suppose that in order for the defendant to be convicted, it takes 8 or more of the 12 jury member to vote that the defendant is guilty. Suppose we assume that the jurors act independently and that each juror makes the right decision with probability .6. What is the probability that the jury makes the right decision? (That is renders a verdict of guilty when the defendant is in fact guilty or renders a verdict of innocent when the defendant is in fact innocent) Assume that with probability .85 the defendant is guilty.

Explanation / Answer

Probability that the defendant is guilty =0.85

Probability that the defendant is not guilty = (1- 0.85) = 0.15

Our required probability is that the jury makes right decision i.ewhen (s)he is guilty 8 or 9 or 10 or 11 or 12 members say that(s)he is guilty.

This is a binomial function with prob of success = 0.6

P(8) = 12C8 * 0.6 ^ 8 * 0.4 ^ (12-8) = 0.2128

P(9) = 12C8 * 0.6^9 * 0.4 ^ 3 = 0.1419

P(10) = 12C10 * 0.6^10 * 0.4^ 2 = 0.0638

P(11) = 12C11 * 0.6^11*0.4^1 = 0.0174

P(12) = 12C12 * 0.6^12 * 0.4^0 = 0.0022

Summing these together we get 0.4382

P(right decision when (s)he is guilty ) = 0.85* 0.4382

P(right decision when (s)he is not guilty ) = 0.15* 0.4382

Therefore not dependent on the probability of being guilty theanswer is (0.85+0.15)*0.4382 = 0.4382. This is the interesting partof the problem

hope it helps. Feel free to ask for clarification

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