A court is investigating the possible occurrence of an unlikely event T. The rel

Question

A court is investigating the possible occurrence of an unlikely event T. The reliability of two independent witnesses Alice and Bob is known to the court: Alice and Bob tel Let A and B be the events that Alice and Bob assert (respectively) that T occurred, and let be the probability that T occurred. (a) What is the probability that T occurred given that both Alice and Bob declare that T occurred? (b) Evaluate the probability that T occurred given that both Alice and Bob declare that T occurred when p-0.9 and = 0.001.

Explanation / Answer

(a) As A and B are both independent events so

Pr(A and B) = Pr(A) * Pr(B)

Pr(Alice and Bob declare that T occured) = Pr(Alice and Bob speaking truth) * Pr(T occured) + Pr(Alice and BOb speaking False) * Pr(T doesn't occured) = p* p * + (1-p) * (1-p) * (1 - ) = p2 + (1-p)2(1-) =

Pr(T occured given that both Alice and Bob declare that T occured) = Pr(T occured l Alice and Bob declare that T occured) = p2 /[p2 + (1-p)2(1-)]

(b) Pr(T occured given that both Alice and Bob declare that T occured) = Pr(T occured l Alice and Bob declare that T occured) = p2 /[p2 + (1-p)2(1-)]

= 0.9 * 0.9 * 0.001/ (0.9 * 0.9 * 0.001 + 0.1 * 0.1 * 0.999) = 0.075

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