In a given city the juries convict guilty defendants 85 of t

In a given city, the juries convict guilty defendants 85% of the time, and acquit innocent ones 91% of the time. In addition, 85% of defendants brought to trial are actually guilty.

a) What is the probability that a random defendant will be guilty but acquitted? (Round your answer to four decimal places.)

b) What is the overall probability that a random defendant will be acquitted? (Round your answer to four decimal places.)

c) If a defendant is acquitted, what is the (conditional) probability that he was actually guilty? (Round your answer to four decimal places.)

Solution

let A be the event that a randomly selected defendant is guilty. then A^c is the event that a randomly selected defendant is innocent.

so P[A]=0.85   and P[A^c]=0.15

let B be the event that the defendant is convicted. so B^c is the event that he is acquitted.

it is given that P[B|A]=0.85    and P[B^c|A^c]=0.91

so P[B^c|A]=0.15 and P[B|A^c]=0.09

a) P[randomly selected defendant will be guilty but acquitted]=P[B^c|A]=0.1500 [answer]

b) P[random defendant will be acquitted]=P[B^c]=P[B^c|A]*P[A]+P[B^c|A^c]*P[A^c] [by total probability theorem]

                                                                    =0.15*0.85+0.91*0.15=0.2640 [answer]

c) P[given that a defendant is acquitted,he was guilty]=P[A|B^c]

now, P[A|B^c]=P[B^c|A]*P[A]/P[B^c] [by bayes\' theorem]

                  =0.15*0.85/0.2640=0.4830 [answer]