Define conditional probability, State and  prove multiplication theorem

Conditional Probability: If $A \& B$  are any two events in a sample space $S. \& P\left(B\right)\ne 0$ . The probability of A given the event B has  occurred is called the conditional probability of $A$ given $B$and is denoted by $P(A/B)$and defined by $P(A/B)=\frac{P(A\cap B)}{P\left(B\right)},P\left(B\right)\ne 0$

Multiplication theorem :

Statement : If A and B are any two events of a random experiment where $P\left(A\right)\ne 0$and $P\left(B\right)\ne 0$then  

$P(A\cap B)=P\left(A\right)P(B/A)=P\left(B\right)P(A/B)$

Proof : Let S be the sample space of the random experiment.

Here A,B are events of the experiment such that $P\left(A\right)\ne 0,P\left(B\right)\ne 0$

According to the definition of conditional probability

$P\left(\frac{B}{A}\right)=\frac{n(B\cap A)}{n\left(A\right)}=\frac{n(B\cap A)}{\frac{n\left(S\right)}{\frac{n\left(A\right)}{n\left(S\right)}}}=\frac{P(B\cap A)}{P\left(A\right)}$

$\therefore P(B\cap A)=P(A\cap B)=P\left(A\right)P(B/A)$

Similarly we can prove

$P(A\cap B)=P\left(B\right)P(A/B)$