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

a) P (residence burglary) = P(burglary of a residence by Alex) or P(burglary of a residence by Becky)

or   P(burglary of a  residence by Carl)

= 0.55 * 0.33 + 0.31* 0.50 + 0.14*1

= 0.4765

note :P(burglary by Alex) = 0.55, P ( residence burglary I Alex) = 0.33

P(burglary by Becky) = 0.31, P ( residence burglary I Becky) = 0.50

P(burglary by Carl) = 1-0.55-0.31 = 0.14 ,  P ( residence burglary I Carl) = 1

b) P (Becky I residence burglary) = P( residence burglary I Becky) .P(Becky) / P (residence burglary)

= 0.50 * 0.31 / 0.4765 = 0.3253