91Ó°ÊÓ

A is an event such that Pr(A) = 0, and B is any other event.

Two events A and B, are said to be independent if the following relation is satisfied:

\({\bf{P}}\left( {{\bf{A}} \cap {\bf{B}}} \right){\bf{ = P}}\left( {\bf{A}} \right) \times {\bf{P}}\left( {\bf{B}} \right)\)- (1)

Since,\(\left( {A \cap B} \right) \subseteq A\); therefore,

\(P\left( {A \cap B} \right) \le P\left( A \right)\)

Using the above equation, we can write:

\(\begin{aligned}{l}P\left( {A \cap B} \right) \le P\left( A \right) &= 0\\ \Rightarrow P\left( {A \cap B} \right) &= 0\end{aligned}\)

Considering the right-hand side of equation (1),

\(\begin{aligned}{c}RHS &= P\left( A \right) \times P\left( B \right)\\ &= 0 \times P\left( B \right)\\ &= 0\\ &= P\left( {A \cap B} \right)\\ &= LHS\end{aligned}\)

Since the relation (1) is satisfied; therefore, it indicates that events A and B are independent events.