P(A B)=16, P(A B)=14, P( A)=14.

Question

$P(A\cup B)=\dfrac16$, $P(A\cap B)=\dfrac14$, $P(\overline A)=\dfrac14$. Determine the relationship of $A,B$.

Accepted Answer

$P(\overline{A\cup B})=\dfrac16\Rightarrow P(A\cup B)=\dfrac56$; $P(\overline A)=\dfrac14\Rightarrow P(A)=\dfrac34$.

$$P(B) = P(A\cup B)-P(A)+P(A\cap B) = \frac56-\frac34+\frac14=\frac13$$

$P(A)\ne P(B)$, but $P(A)P(B)=\dfrac34\times\dfrac13=\dfrac14=P(A\cap B)$, so independent but not equally likely.

Solutions · 1