Question

Problem 30
An individual experiences a loss due to property damage and a loss due to
bodily injury. Losses are independent and uniformly distributed on the in-
interval [0,3]. Calculate the expected loss due to bodily injury, given that at
least one of the losses is less than 1.
(A) 0.50
(B) 1.00
(C) 1.10
(D) 1.25
(E) 1.50

The answer is 1.10

Answer 1

P(at least one of losses are less than 1)=P(property damage less than 1 and bodily injury higher than 1)+P(property damage higher than 1 and bodily injury less than 1)+P(property damage and bodily injury less than 1)

=(1/3)*(2/3)+(1/3)*(2/3)+(1/3)*(1/3)=5/9

therefore P(bodily injury less than 1|at least one of losses are less than 1)

=(P(property damage less than 1 and bodily injury higher than 1)+P(property damage and bodily injury less than 1))/P(at least one of losses are less than 1) =((1/3)*(2/3)(1/3)*(1/3))/(5/9)=3/5

and P(bodily injury greater than 1|at least one of losses are less than 1)=1-3/5=2/5

here expected loss if it is less than 1 =(0+1)/2=0.5

and  expected loss if it is more than 1=(1+3)/2=2

therefore Expected loss due to bodily injury, given that at
least one of the losses is less than 1 =xP(x)=(3/5)*(0.5)+(2/5)*(2)=1.1