P(A/B) is known as conditional probability and it means the probability of event A that depends on another event B and is read as "probability of A given B". It saysP(A/B) = P(A∩B) / P(B).
It is also known as "the probability of A given B". P(A/B) Formula is used to find this conditional probability quickly.
What is P(A/B) Formula?
The conditional probability P(A/B) arises only in the case of dependent events. It gives the probability of A given that B has occurred.
P(A/B) Formula
The probability of A given B formula says:
P(A/B) = P(A∩B) / P(B)
(Similarly, the P(B/A) formula is: P(B/A) = P(A∩B) / P(A))
Here,
- P(A) = Probability of event A happening.
- P(B) = Probability of event B happening.
- P(A∩B) = Probability of happening of both A and B.
From these two formulas, we can derive the product formulas of probability.
- P(A∩B) = P(A/B) × P(B)
- P(A∩B) = P(B/A) × P(A)
☛Note: If A and B are independent events, then P(A/B) = P(A) or P(B/A) = P(B) and in this case, the above two formulas together turn intoP(A∩B) = P(A) × P(B). This is referred to as the condition for two events to be independent.
Have questions on basic mathematical concepts?
Become a problem-solving champ using logic, not rules. Learn the why behind math with our certified experts
Book a Free Trial Class
P(A/B) Formula Examples
Example 1: When a fair die is rolled, what is the probability of A given B where A is the event of getting an odd number and B is the event of getting a number less than or equal to 3?
Solution:
To find: P(A/B) using the given information.
When a die is rolled, the sample space = {1, 2, 3, 4, 5, 6}.
A is the event of getting an odd number. So A = {1, 3, 5}.
B is the event of getting a number less than or equal to 3. So B = {1, 2, 3}.
Then A∩B = {1, 3}.
Using the P(A/B) formula:
P(A/B) = P(A∩B) / P(B)
\(P(A/B) = \dfrac{2/6}{3/6} = \dfrac 2 3\)
Answer: ∴ P(A/B) = 2 / 3.
Example 2: Two cards are drawn from a deck of 52 cards where the first card is NOT replaced before drawing the second card. What is the probability that both cards are kings?
Solution:
To find: The probability that both cards are kings.
P(card 1 is a king) = 4 / 52 (as there are 4 kings out of 52 cards).
P(card 2 is a king/card 1 is a king) = 3 / 51 (as the first king is not replaced, there is a total of 3 kings out of 51 left out cards).
By the formula of conditional probability,
P(card 1 is a king ∩ card 2 is a king) = P(card 2 is a king/card 1 is a king) × P(card 1 is a king)
P(card 1 is a king ∩ card 2 is a king) = 3 / 51 × 4 / 52 = 1 / 221
Answer: The required probability = 1 / 221.
Example 3:What is the probability that a selected person is a smoker given it is male?
| Male | Female | Total | |
|---|---|---|---|
| Smoker | 45 | 25 | 70 |
| Nonsmoker | 75 | 55 | 130 |
| Total | 120 | 80 | 200 |
Solution:
We have to find P(smoker | male)
By using the probability of A given B formula, P(A | B) =P(A∩B) / P(B).
Using this, we can write
P(smoker | male) = P (smoker∩ male) / P(male)
= (45/200) / (120/200)
= 45/120
= 9/24
Answer:∴ The required probability = 9/24.
FAQs on P(A/B) Formula
What is the Probability of A Given BFormula?
The probability of A given B formula is used to calculate the conditional probability such that we have to find the probability of event 'A' occurring which happens after event 'B' has occurred. P(A/B) formula is given as, P(A/B) = P(A∩B) / P(B), where, P(A) is the probability of the event A,P(B) is the probability of the event B,and P(A∩B) is the probability of happening of both A and B.
How to Find P(A∩B) using P(A/B) Formula?
P(A∩B) can be calculated using the P(A/B) Formula as, P(A∩B) = P(A/B) × P(B), where, P(B) is the probability of happening of event B and P(A∩B) is the probability of A and B.
If A and B are Independent Events then What is the Condition?
If A and B are independent eventsthen there is no question of conditional probability. i.e., P(A/B) is just P(A) and P(B/A) is just P(B). Thus, the probability of A and B in this case is just the product of individual probabilities. i.e., P(A and B) = P(A)· P(B).
What is ∩ Symbol in P(A∩B) Formula?
The probability of A given B formula is given as, P(A/B) = P(A∩B) / P(B), here ∩ symbol represents the intersection of event 'A' and event 'B'. Thus, P(A∩B) represents the probability of happening of both A and B together.
What is P(A∩B) Formula?
P(A∩B) is the probability of both independent events “A” and "B" happening together, P(A∩B) formula can be written as
- P(A∩B) = P(A) × P(B), where A and B are independent
- P(A∩B) = P(A/B) × P(B) (or) P(B/A)× P(A), where A and B are dependent.
