A person tried by a 3 judge panel is declared guilty if at...

A person tried by a 3-judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability 0.7, whereas when the defendant is in fact innocent, this probability drops to 0.2. If 70 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that (a) judges 1 and 2 vote guilty; (b) judges 1 and 2 cast 1 guilty and 1 not guilty vote; (c) judges 1 and 2 both cast not guilty votes. Let Ei , i = 1, 2, 3 denote the event that judge i casts a guilty vote. Are these events independent? Are they conditionally independent? Explain.

Transcript

00:01 So i'm going to say off the bat here that we have the events, j .i equals g or n, indicating the ith judge votes guilty or not guilty.

00:23 So for part a, we are looking for probability that j3 equals g given j1 equals g and j2 equals g.

00:42 So what we need to do to find this is we basically need to sum over the probabilities where we could have that the defendant or the accused is either guilty or not guilty.

00:55 So if we assume that the defendant is guilty, then we'd have 0 .7 times 0 .7 times 0 .7.

01:05 That's for the probability of each judge voting guilty.

01:08 So we have j1 equals g, j2 equals g, j3 equals g, and then we need to multiply that by the probability that the defendant is actually guilty, which we are told is 0 .7.

01:26 Then we need to add on the probability that j1 votes g and j2 votes g and j3 votes g, given that the defendant is actually...

01:40 Oh, actually, pardon me, i just realized i'm making a slight mistake there.

01:44 Those probabilities should be 0 .2, given that the defendant is actually not guilty.

01:51 We know that the defendant being not guilty occurs with probability 0 .3.

01:56 And so when we plug all of these values in, we'll find a result of 0 .24 .25.

02:03 Then for part b, the probability of...

02:09 Actually, i'll just refer to the probability that we're looking for in part b as just that will be equal to the probability that j3 equals g given j1 equals g j2 equals n plus the probability that j3 equals g given j1 equals n plus j2 equals g or not plus but uh and j2 equals g so we'd have that this is going to turn out to be 0 .7 cubed times 0 .4...

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