Question

Two events E1 and E2 associated with a sample space of an experiment are such that P(E1)=0.27, P(E2)=0.40 and P(E1 U E2)=0.58. Determine whether E1 and E2 are independent.

Answer 1

$$P(E_1\cup E_2)=P(E_1)+P(E_2)-P(E_1\cap E_2)$$

$$P(E_1\cap E_2)=P(E_1)+P(E_2)-P(E_1\cup E_2)$$

$$=0.27+0.40-0.58=0.09$$

$$P(E_1)P(E_2)=0.27(0.40)=0.108$$

$$\not=0.09=P(E_1\cap E_2)$$

Since $P(E_1\cap E_2)\not=P(E_1)P(E_2),$ then $E_1$ and $E_2$ are not independent.