P a B Conditional Probability

Conditional Probability Definition:-
If B is any arbitrary event in a sample space and P ( B ) > 0. The probability of A given B is defined by
P ( A / B ) = P ( A n B ) / P ( B ).
This is called conditional probability of A given B and is written as P ( A / B ).

Proof of Conditional Probability
Let A and B be any two events in a random experiment associated with the sample space "S" then
if n ( S ) = number of elements in S
n ( B ) = number of elements in B
n ( A n B ) = number of elements in A n B.
P ( B ) = n ( B ) / n ( S )
P ( A n B ) = n ( A n B ) / n ( S )

Now for the conditional event A after B the favorable outcomes must be one of the sample points "B"
In other words, the sample space for the event A after B is B.
P ( A / B ) = n ( A ) / n ( B )
P ( A / B ) =n ( A n B ) / n ( B )                [ A n B = A ]
P ( A / B ) =  [ P ( A n B ) / n ( S ) ] / [ P ( B ) / n ( S ) ]
P ( A / B ) = P ( A n B ) / P ( B )
 Hence Proved.

Example of Conditional Probability
If the probability that a communication system will have fidelity is 81 and the probability that it will have fidelity and selectivity is 18. What is the probability that a system will high fidelity will also have high selectivity?
Solution:-
Let the event B will be the communication system to have high fidelity
P ( B ) = 81
Let the event to have high fidelity and selectivity is A n B
P ( A n B ) = 18
By the definition of Conditional probability
P (A / B) = P (A n B) / P (B)
Probability to have selectivity if it will have fidelity
P (A / B) = 18 / 81
P (A / B) = 2 / 9  = 0.22.....

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