P(A/B) Formula
P(A/B) is known as conditional probability and it means the probability of event A that depends on another event 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 conditional probability of A given that B has occurred.
P(A/B) Formula
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)
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.
FAQs on P(A/B) Formula
What is P(A/B) Formula?
P(A/B) Formula is the formula used to calculate the conditional probability such that we have to find the probability of event 'A' occurring when event 'B' has occurred. P(A/B) Formula is given as, P(A/B) = P(A∩B) / P(B), where, P(A) is probability of event A happening, P(B) is the probability of event B happening 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 event B happening and P(A∩B) is the probability of happening of both A and B.
What is ∩ Symbol in P(A∩B) Formula?
P(A/B) Formula is given as, P(A/B) = P(A∩B) / P(B), here ∩ symbol represents the intersection of event 'A' and event 'B'. P(A) is probability of event A happening, P(B) is the probability of event B happening and P(A∩B) is the probability of happening of both A and B.
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,
- P(A∩B) = Probability of both independent events “A” and "B" happening together.
- P(A) = Probability of an event “A”
- P(B) = Probability of an event “B”