91Ó°ÊÓ

S is the sample space of an experiment and A is any event in that space.

The conditional probability of the event A given that the sample space S has previously occurred is given by:

\(\Pr \left( {A\left| S \right.} \right) = \frac{{\Pr \left( {A \cap S} \right)}}{{\Pr \left( S \right)}}\;\;\;\;\;\;\;\;\;\;\;\;\;...\left( 1 \right)\)

Since,A is any event in the sample space S; therefore,

\(\left( {A \cap S} \right) = A\)

This implies:

\(\Pr \left( {A \cap S} \right) = \Pr \left( A \right)\)

Also, the probability of sample space is 1.

Mathematically, \(\Pr \left( S \right) = 1\)

Thus, equation (1) becomes:

\(\begin{aligned}{}\Pr \left( {A\left| S \right.} \right) &= \frac{{\Pr \left( {A \cap S} \right)}}{{\Pr \left( S \right)}}\\ &= \frac{{\Pr \left( A \right)}}{1}\\ &= \Pr \left( A \right)\end{aligned}\)

Therefore, if S is the sample space of an experiment and A is any event in that space, then the value of \(\Pr \left( {A\left| S \right.} \right)\) is \(\Pr \left( A \right)\).