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) or P(B/A) arises only in the case of dependent events.

  • The P(A/B) formula is:

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∩B) is the 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)

P(A/B) formula of conditional probability

Note: If A and B are independent events, then 

P(A/B) = P(A)

P(B/A) = P(B)

Solved Examples Using P(A/B) Formula

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.