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 .7, whereas when the defendant is in fact innocent, this probability drops to .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.