In a certain small town, 3 professional burglars are currently out of prison: Alex, Becky, and Carl. Alex has in the past committed 55% of the burglaries committed by the three, Becky 31%, and Carl the rest. But only 1/3 of Alex's jobs are burglaries of a residence, while half of Becky's are, and all of Carl's are. a) What is the probability that the next burglary in town (if one of the three did it) is the burglary of a residence? b) Sure enough, a resident reports a home burglary. If one of the three did it, what is the probability Becky was guilty?

Answer explained below

Step-by-step explanation:

Alex has in the past committed 55% of the burglaries committed by the three, Becky 31%, and Carl the rest, hence

probability, P (Carl doing a burglary) = 1 - P (Alex or Becky doing a burglary)

= 1 - (0.55 + 0.31)

= 0.14

a) P (next burglary in town is the burglary of a residence) = P (Alex did a burglary of a residence) + P (Becky did a burglary of a residence) + P (Carl did a burglary of residence

= 0.55*1/3 + 0.31*0.5 + 0.14*1

= 0.4783

b) From Bayes' Theorem: P (A | B) = P (A & B) / P (B)

Hence,

P (Becky | a resident reports a home burglary) = P (Becky did a burglary of a residence) / P (burglary of a residence)

= (0.31 * 0.5) / 0.4783

= 0.3240