 # Conditional Probability Formula

Conditional probability formula gives the measure of the probability of an event given that another event has occurred. If the event of interest is A and the event B is known or assumed to have occurred, “the conditional probability of A given B”, or “the probability of A under the condition B”. The events are usually written as P(A|B), or sometimes P B(A). The formula for conditional probability for both the conditions i.e. “the probability of A under the condition B” and “the probability of B under the condition A” are stated below.

## Formula for Conditional Probability

 Conditional Probability of A given B P (A|B) = P(A ∩ B)⁄P(B) Conditional Probability of B given A P (B|A) = P(B ∩ A)⁄P(A)

### Solved Examples Using Conditional Probability Formula

Question 1:

The probability that it is Friday and that a student is absent is 0.03. Since there are 5 school days in a week, the probability that it is Friday is 0.2. What is the probability that a student is absent given that today is Friday?

Solution:

The formula of Conditional probability Formula is:

P (B|A) = P(A ∩ B)⁄P(A)

P(Absent | Friday)= P (Absent and Friday)⁄P(Friday)

= 0.03/0.2

= 0.15

= 15 %

Question 2: A teacher gave her students of the class two tests namely maths and science. 25% of the students passed both the tests and 40% of the students passed the maths test. What percent of those who passed the maths test also passed the science test?

Solution:
Given,
Percentage of students who passed the maths test = 40%
Percentage of students who passed both the tests = 25%
Let A and B be the events of the number of students who passed maths and science tests.
According to the given,
P(A) = 40% = 0.40
P(A ⋂ B) = 25% = 0.25
Percent of students who passed the maths test also passed the science test
= Condition probability of B given A
= P(B|A)
= P(A ⋂ B)/P(A)
= 0.25/0.40
= 0.625
= 62.5%

Question 3: A bag contains green and yellow balls. Two balls are drawn without replacement. The probability of selecting a green ball and then a yellow ball is 0.28. The probability of selecting a green ball on the first draw is 0.5. Find the probability of selecting a yellow ball on the second draw, given that the first ball drawn was green.

Solution:
Let A and B be the events of drawing a green in the first draw and yellow ball in the second draw respectively.
From the given,
P(A) = 0.5
P(A ⋂ B) = 0.28
Probability of selecting a yellow ball on the second draw, given that the first ball drawn was green = Conditional of B given A
= P(B|A)
= P(A ⋂ B)/P(A)
= 0.28/0.5
= 0.56