Let E_1,E_2 be events with P(E_1 E_2)=12, P(E_2 E_1)=34, P(E_1 E_2)=18.

Question

Let $E_1,E_2$ be events with $P(E_1\mid E_2)=\dfrac12$, $P(E_2\mid E_1)=\dfrac34$, $P(E_1\cap E_2)=\dfrac18$. Then:

Accepted Answer

From $P(E_1\mid E_2)=\dfrac{P(E_1\cap E_2)}{P(E_2)}=\dfrac12\Rightarrow P(E_2)=\dfrac14$. From $P(E_2\mid E_1)=\dfrac34\Rightarrow P(E_1)=\dfrac{1/8}{3/4}=\dfrac16$.

Testing option C: $P(E_1\cap E_2')=P(E_1)-P(E_1\cap E_2)=\dfrac16-\dfrac18=\dfrac{1}{24}$, and $P(E_1)P(E_2)=\dfrac16	imes\dfrac14=\dfrac{1}{24}$ --- equal, confirming option C.

Solutions · 1