The probability of an event B occurring when it is known that some event A has occurred is called as the conditional probability of event B. It is denoted by P(B/A). It is also called as the probability of B, given A.
We need to find the probability of B relative to the sample space for A, which is a reduced sample space. The conditional probability P(B/A) can be defined as
P(B/A) = P(A ∩ B) / P(A) , where P(A ∩ B) & P(A) are found from the original sample space, S.
Example:- Let B be an event of getting a perfect square when a die is tossed. And A be the event that the number on the dice is an odd number. Find P(B/A).
Answer:- sample space, S = {1,2,3,4,5,6} and A = {1,3,5} and B = {1,4}
P(A) = 3/6 = 1/2. And P(B) = 2/6 = 1/3. And P(A ∩ B) = 1/6.
Now P(B/A) = P(A ∩ B) / P(A) = (1/6) / (1/2) = 2/6 = 1/3.
The conditional probability formula can be re-written as
P(A ∩ B) = P(B/A).P(A) &
P(B ∩ A) = P(A/B).P(B)
These are also known as the product rules. Also,
P(A ∩ B) = P(B ∩ A)
P(B/A).P(A) = P(A/B).P(B)
Two cards are drawn from a deck of cards. If the first card is an ace of spades, find the probability that the second card is a ace.
A pair of dice are thrown. Given that the first dice shows 4, find out the probability that the total of both the dice is greater than 7.
One bag contains 4 red balls and 3 black balls, and the second bag contains 3 white balls and 7 red balls. Exactly one ball is drawn from the second bag at random and placed in the first bag. What is the probability that the ball now drawn from the first bag is a red ball?
Two cards are drawn at random from deck of cards. If the first card is a red card, find the probability that the second card is a Jack.