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