In some military courts, 9 judges are appointed. A defendant is declared guilty

Question

In some military courts, 9 judges are appointed. A defendant is declared guilty if the majority of judges cast votes of guilty, and the defendant is declared innocent otherwise. 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 0.3.

(a) Given that the defendant is in fact guilty, find the probability that this defendant is declared guilty.

(b) Given that the defendant is in fact innocent, find the probability that this defendant is declared guilty.

(c) Assume that 60% defendants are, in fact, guilty. Find the probability that a defendant is in fact guilty, when it is declared guilty.

Explanation / Answer

P(judging guilty/Guilty) = 0.7

P(judging innocent/Guilty) = 1-0.7 = 0.3

P(Judging guilty/Innocent) = 0.3

P(Judging Innocent/Innocent) = 1-0.3 = 0.7

a)probability that this defendant is declared guilty given that the defendant is in fact guilty

= P(5 judges votes guilty/Guilty)+P(6 judges votes guilty/Guilty)+P(7 judges votes guilty/Guilty)+P(8 judges votes guilty/Guilty)+P(9judges votes guilty/Guilty)

= (9C5)(0.7)5(0.3)4+(9C6)(0.7)6(0.3)3+(9C7)(0.7)7(0.3)2+(9C8)(0.7)8(0.3)1+(9C9)(0.7)9 = 0.9011

b)probability that this defendant is declared guilty given that the defendant is in fact innocent

= P(5 judges votes guilty/innocent)+P(6 judges votes guilty/innocent)+P(7 judges votes guilty/innocent)+P(8 judges votes guilty/innocent)+P(9judges votes guilty/innocent)

= (9C5)(0.3)5(0.7)4+(9C6)(0.3)6(0.7)3+(9C7)(0.3)7(0.7)2+(9C8)(0.3)8(0.7)1+(9C9)(0.3)9 = 0.0988

c)

P(defendant is guilty) = 0.6

P(defendant is innocent) = 0.4

P(defendant is decalred guilty) = 0.6*0.9011+0.4*0.0988 = 0.58018

probability that a defendant is in fact guilty, when it is declared guilty = P(defendant is in fact guilty,/declared guilty)

= P(defendant is in fact guilty and declared guilty)/P(declared guilty) = 0.6*(0.9011)/0.58018 = 0.93188

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