Let ${E_1}$ and ${E_2}$ be two independent events such that $P\left( {{E_1}} \right) = {P_1}$ and $P\left( {{E_2}} \right) = {P_2}$. Describe in words of the events whose probabilities are

(i) ${P_1}{P_2}$

(ii) $\left( {1 - {P_1}} \right){P_2}$

(iii) $1 - \left( {1 - {P_1}} \right)\left( {1 - {P_2}} \right)$

(iv) ${P_1} + {P_2} - 2{P_1}{P_2}$

$P\left( {{E_1}} \right) = {P_1}$ and $P\left( {{E_2}} \right) = {P_2}$

(i) ${P_1}{P_2} \Rightarrow P\left( {{E_1}} \right) \cdot P\left( {{E_2}} \right) = P\left( {{E_1} \cap {E_2}} \right)$

So, ${E_1}$ and ${E_2}$ occur.
(ii) $\left( {1 - {P_1}} \right){P_2} = P{\left( {{E_1}} \right)^\prime } \cdot P\left( {{E_2}} \right) = P\left( {E_1^\prime \cap {E_2}} \right)$

So, ${E_1}$ does not occur but ${E_2}$ occurs.

(iii) $1 - \left( {1 - {P_1}} \right)\left( {1 - {P_2}} \right) = 1 - P{\left( {{E_1}} \right)^\prime }P{\left( {{E_2}} \right)^\prime } = 1 - P\left( {E_1^\prime \cap E_2^\prime } \right)$

$= 1 - \left[ {1 - P\left( {{E_1} \cup {E_2}} \right)} \right] = P\left( {{E_1} \cup {E_2}} \right)$

So, either ${E_1}$ or ${E_2}$ or both ${E_1}$

and ${E_2}$ occurs.

(iv) ${P_1} + {P_2} - 2{P_1}{P_2} = P\left( {{E_1}} \right) + P\left( {{E_2}} \right) - 2P\left( {{E_1}} \right) \cdot P\left( {{E_2}} \right)$

$= P\left( {{E_1}} \right) + P\left( {{E_2}} \right) - 2P\left( {{E_1} \cap {E_2}} \right)$

$= P\left( {{E_1} \cup {E_2}} \right) - P\left( {{E_1} \cap {E_2}} \right)$

So, either ${E_1}$ or ${E_2}$ occurs but not both.