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 and is read as "probability of A given B". It says P(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 into P(A∩B) = P(A) × P(B). This is referred to as the condition for two events to be independent.

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?

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.