Independent solution

How to solve this Law of Total Probability question

Setup

Setup

Condition on the claim-count state. The interval is impossible in the zero-claim state, so only the one-claim and multiple-claim states contribute.

P(N=1)=13,P(N>1)=16P(N=1)=\frac13,\qquad P(N>1)=\frac16

Model

Model

For each contributing state, compute the exponential probability of falling between four and eight, then weight it by that state's probability.

P(4<S<8)=13(e−4/5−e−8/5)+16(e−4/8−e−8/8)P(4<S<8)=\frac13(e^{-4/5}-e^{-8/5})+\frac16(e^{-4/8}-e^{-8/8})

Compute

Compute

Adding the two weighted interval probabilities gives approximately 0.12225.

P(4<S<8)=0.1220P(4<S<8)=0.1220

Answer

Answer

The unconditional interval probability is approximately 0.12, selecting choice C.

0.12(C)\boxed{0.12\quad\text{(C)}}