Dans Diner employs three dishwashers Al washes 80 of the dis

Dan\'s Diner employs three dishwashers. Al washes 80% of the dishes and breaks only 1% of those he handles. Betty and Chuck each wash 10% of the dishes, and Betty breaks only 1% of hers, but Chuck breaks 5% of the dishes he washes. You go to Dan\'s for supper one night and hear a dish break at the sink. What\'s the probability that Chuck is on the job?

Solution

Let A1,A2,A3 be the event that AI,Betty, Chucks is on job respectively.

Let B be the event that a dish breaks.

P(A1)=0.8,P(A2)=0.1,P(A3)=0.1

P(B|A1)=0.01,P(B|A2)=0.01,P(B|A3)=0.05

P(A3|B)=P(A3)P(B|A3)/{P(A1)P(B|A1)+P(A2)P(B|A2)+P(A3)P(B|A3)}

=5/14=0.357