Independent solution

How to solve this Conditional Probability question

Setup

Setup

Let H denote the higher claimed amount and O denote an overstated amount. Record the supplied marginals and intersection.

Pr⁡(H)=0.70,Pr⁡(O)=0.50,Pr⁡(H∩O)=0.45\Pr(H)=0.70,\qquad \Pr(O)=0.50,\qquad \Pr(H\cap O)=0.45

Model

Model

Remove the overlap with H from the overstatement marginal, and take the complement of H for the conditioning event.

Pr⁡(O∩Hc)=Pr⁡(O)−Pr⁡(O∩H)=0.50−0.45=0.05\Pr(O\cap H^c)=\Pr(O)-\Pr(O\cap H)=0.50-0.45=0.05
Pr⁡(Hc)=1−0.70=0.30\Pr(H^c)=1-0.70=0.30

Compute

Compute

Normalize the joint mass by the probability of the conditioning group.

Pr⁡(O∣Hc)=Pr⁡(O∩Hc)Pr⁡(Hc)\Pr(O\mid H^c)=\frac{\Pr(O\cap H^c)}{\Pr(H^c)}
Pr⁡(O∣Hc)=0.050.30=16\Pr(O\mid H^c)=\frac{0.05}{0.30}=\frac16

Answer

Answer

The required conditional probability is one sixth.

16(C)\boxed{\frac16\quad\text{(C)}}