Conditional probability

Definition :

The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occured. This probability is written P(B|A) , notation for the probability of B given A.

In the case where events A and B are independent ( where event A has no effect on the probability of event B ), the conditional probability of event B given event A is simply the probability of event B, that is P(B).

If events A and B are not independent, then the probability of the intersection of A and B ( the probability that both events occur ) is defined by P(A ∩ B) = P(A) . P(B|A).

From this definition, the conditional probability P(B|A) is easily obtained by dividing by P(A):

Now, to calculate the probability of the intersection of more than two events, the conditional probabilities of all the preceding events must be considered. For example in the case of three events, A, B and C, the probability of the intersection :

P( A ∩ B ∩C )  = P(A) . P(B|A) . P(C|A ∩ B)

Another important method for calculating conditional probabilities is given by Bayes's formula. The formula is based on the expression :

P( B ) = P ( B|A ) P( A ) + P( B|A^c ) P( A^c )

which simply states that the probability of event B is the sum of the conditional probabilities of event B given that event A has or has not occured. For independent events A and B, this is equal to :

P( B ). P( A ) + P( B ). P( A^c )  = P( B ) ( P( A ) + P( A^c ) ) = P( B ) ( 1 ) = P( B )

Since the probability of an event and its complement must always sum to 1.

Bayes Formula :