Independent solution

How to solve this Conditional Probability question

Setup

Setup

Let M and D denote the two insurance events and express every probability relative to the size of M.

Pr⁡(D∣M)=0.25,Pr⁡(M∣D)=0.40\Pr(D\mid M)=0.25,\qquad \Pr(M\mid D)=0.40

Model

Model

Both conditional statements describe the same overlap, so equate their two representations.

Pr⁡(M∩D)=0.25Pr⁡(M)=0.40Pr⁡(D)\Pr(M\cap D)=0.25\Pr(M)=0.40\Pr(D)
Pr⁡(D)Pr⁡(M)=0.250.40=0.625\frac{\Pr(D)}{\Pr(M)}=\frac{0.25}{0.40}=0.625

Compute

Compute

Apply inclusion–exclusion and divide through by the medical-insurance probability.

Pr⁡(M∪D)Pr⁡(M)=1+Pr⁡(D)Pr⁡(M)−Pr⁡(M∩D)Pr⁡(M)\frac{\Pr(M\cup D)}{\Pr(M)}=1+\frac{\Pr(D)}{\Pr(M)}-\frac{\Pr(M\cap D)}{\Pr(M)}
=1+0.625−0.25=1.375=1+0.625-0.25=1.375

Answer

Answer

The ratio is the value shown in choice A.

1.375(A)\boxed{1.375\quad\text{(A)}}