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Multiplication Theorems on Probability

TopProbability is the branch of mathematics which deals with the occurrence of samples. The basic form of Multiplication theorems on probability for two events ‘X’ and ‘Y’ can be stated as,
p ( x . y ) = p ( x ) . p ( x / y ).
Here p ( x ) and p ( y ) are the Probabilities of occurrences of events ‘x’ and ‘y’ respectively.
P ( x / y ) is the Conditional Probability of ‘x’ and the condition is that ‘y’ has occurred before ‘x’.
P ( x / y ) is always calculated after ‘y’ has occurred. Here, occurrence of ‘x’ depends on ‘y’.
‘y’ has changed some events already. So, occurrence of ‘x’ also changes.
The essential condition is that ‘y’ is not equals to zero that is y ≠ 0.
Now, consider the case when ‘x’ and ‘y’ are Independent Events.
The occurrence of ‘x’ does not depend on ‘y’ as they are independent events.
Hence,
p ( x / y ) = p ( x ) - - - - - - - - - - - - - - - - - - ( equation 1 )
As occurrence of ‘y’ has no effect on ‘x’.
Now, according to the multiplication theorem of probability,
P (x . y ) = p ( x ) . p ( x / y ) - - - - - - - - - - - - - ( equation 2 )
Substituting p ( x / y ) from “equation 2” in “equation 1” , we get
p ( x . y ) = p ( x ) . p ( y ),
This is the special case of this theorem.
This case is valid only when events are independent.