In a jury trial, suppose that inorder for the defendant to be convicted, it take

Question

In a jury trial, suppose that inorder for the defendant to be convicted, it takes 8 or more of the12 jury member to vote that the defendant is guilty. Suppose weassume that the jurors act independently and that each juror makesthe right decision with probability .6. What is the probabilitythat the jury makes the right decision? (That is renders a verdictof guilty when the defendant is in fact guilty or renders a verdictof innocent when the defendant is in fact innocent) Assume thatwith 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 decisioni.e when (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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