# Conditional Probability

Conditional Probability

The probability of B under the assumption that A takes place. It is denoted by P(B/A) and is called the conditional probability of B given that A takes place

Sometimes the probability of a given event depends on the occurrence or non-occurrence of some other event. Suppose A and B are two events in a sample space S.

Let n = number of sample points in S

m₁ = number of sample points in A

m₂ = number of sample points in B

m₁₂ = number of sample points in A∩B.

Then,

P(A) = m₁/ n, P(B) = m₂/n and P(A∩B) =   m₁₂/n

The probability of A∩B, (i.e. of B) in the sample space A is m₁₂/ m₁ .  This is the probability of B under the assumption that A takes place. It is denoted by P (B/A) and is called the conditional probability of B given that A takes place.

Therefore, P(B/A) = m₁₂/ m₁ = n(A∩B)/n(A), provided n(A) ≠ 0.

Similarly, P(A/B) = m₁₂/ m₂ = n(A∩B)/n(B), provided n(B) ≠ 0.

Two events A and B are said to be independent, if P(A/B) = P(A) and P(B/A) = P(B)

Example: What is the probability of rolling dice and it’s value is less then 4

Solution: P(B/A)  = n(A∩B)/n(A), where n(A) ≠ 0

We can use above formula

P(B/A) =⅙/½ = ⅓