Conditional probability is known as the possibility of an event or outcome happening, based on the existence of a previous event or outcome. It is calculated by multiplying the probability of the preceding event by the renewed probability of the succeeding, or conditional, event.
The probability of occurrence of any event A when another event B in relation to A has already occurred is known as conditional probability. It is depicted by P(A|B).
As depicted by the above diagram, sample space is given by S, and there are two events A and B. In a situation where event B has already occurred, then our sample space S naturally gets reduced to B because now the chances of occurrence of an event will lie inside B.
As we have to figure out the chances of occurrence of event A, only a portion common to both A and B is enough to represent the probability of occurrence of A, when B has already occurred. The common portion of the events is depicted by the intersection of both the events A and B, i.e. A ∩ B.
This explains the concept of conditional probability problems, i.e. occurrence of any event when another event in relation to has already occurred.
When the intersection of two events happen, then the formula for conditional probability for the occurrence of two events is given by;
P(A|B) = N(A∩B)/N(B)
Or
P(B|A) = N(A∩B)/N(A)
Where P(A|B) represents the probability of occurrence of A given B has occurred.
N(A ∩ B) is the number of elements common to both A and B.
N(B) is the number of elements in B, and it cannot be equal to zero.
Let N represent the total number of elements in the sample space.
Since N(A ∩ B)/N and N(B)/N denotes the ratio of the number of favourable outcomes to the total number of outcomes; therefore, it indicates the probability
Therefore, N(A ∩ B)/N can be written as P(A ∩ B) and N(B)/N as P(B).
⇒ P(A|B) = P(A ∩ B)/P(B)
Therefore, P(A ∩ B) = P(B) P(A|B) if P(B) ≠ 0
= P(A) P(B|A) if P(A) ≠ 0
Similarly, the probability of occurrence of B when A has already occurred is given by,
P(B|A) = P(B ∩ A)/P(A)
