Question

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 1

Answer explained below

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