Conditional Probability Formula
The concept of the conditional probability formula is primarily related to the Bayes’ theorem, which is one of the most influential theories in statistics. Conditional probability is the probability for one event to occur with some relationship to one or more other events. For example:
 Event A: it's a 0.4 (40%) chance of raining today.
 Event B: I will go outside and it has a probability of 0.5 (50%).
A conditional probability looks at these two events in relationship with one another, the probability that it is both raining and I will go outside. Let us understand the conditional probability formula using solved examples. Please note that conditional probability doesn't state that there is always a causal relationship between the two events, also it does not indicate that both events occur simultaneously.
What Is Conditional Probability Formula?
The concept of the conditional probability formula is one of the quintessential concepts in probability theory.
The Bayes' theorem is used to determine the conditional probability of event A, given that event B has occurred, by knowing the conditional probability of event B, given the event A has occurred, also the individual probabilities of events A and B.
The formula for conditional probability is :
P(A  B) = P(A and B)/P(B)
It can also be written as,
\(P( {AB} ) = \frac{{P( {A \cap B} )}}{{P( B )}}\)
Let us understand the conditional probability formula using solved examples.
Solved Examples Using Conditional Probability Formula

Example 1: In a group of 10 people, 4 people bought apples, 3 bought oranges, and 2 bought apples and oranges. If a buyer chosen at randomly bought apples, using the conditional probability formula find out what is the probability they also bought oranges?
Solution:
Let people who bought apples are A and who bought oranges are O.
It’s given that
P(A) = 4 out of 10 = 40% = 0.4
P(O) = 3 out of 10 = 30% = 0.3Hence,
P(A∩O) = 2 out of 10 = 20% = 0.2Now, using the conditional probability formula
P(OA) = P(A∩O) / P(A) = 0.2 / 0.4 = 0.5 = 50%Answer: The probability that a buyer bought oranges, given that they bought apples, is 50%.

Example 2: My neighbor has 2 children. I learn that she has a son, Adam. What is the probability that Adam’s sibling is a boy?
Solution:
Let the boy child be B and the girl child is G.
The sample space is S = {BB,BG,GB,GG}
Assume that boys and girls are equally likely to be born, the 4 elements of S are equally likely.
The event, X, that the neighbor has a son is the set X = {BB,BG,GB}
Hence, P(X) = 3/4
The event, Y, that the neighbor has two sons is the set Y = {BB}
Then, P(Y ∩ X) = 1/4
Now, using the conditional probability formula
P(Y  X) = P(Y ∩ X) / P(X) = (¼) / (¾) = ⅓Answer: The probability that Adam’s sibling is a boy is 1/3.