The probability of A, given B has occurred is denoted by P(A|B). Conditional probability of an event A is calculated when the event B has already occurred by the given formula.

          $P(A|B)$ = $\frac{P(A and B)}{P(B)}$
Example:

In an exam, two reasoning problems, 1 and 2, are asked. 35% students solved problem 1 and 15% students solved both the problems. How many students who solved the first problem will also solve the second one?

Solution:

Probability of student solving problem $1, P(1) = 0.35$

Probability of student solving both problem, $P(1\ and\ 2) = 0.15$

Probability of solving 2 if 1 is solved, $P(2|1)$ = $\frac{P(1 and 2)}{P(1)}$ = $\frac{0.15}{0.35}$ = $0.428$

Conditional Probability Word Problems

Problem 1:

Out of 50 people surveyed in a study, 35 smoke in which there are 20 males. What is the probability the if the person surveyed is a smoker then he is a male?

Solution:

Probability of the person being male and a smoker, $P(A\ and\ B)$ = $\frac{20}{50}$

Probability of person being smoker, $P(A)$ = $\frac{35}{50}$

Probability of a person being male if he is smoker, $P(B|A)$ = $\frac{P(A and B)}{P(A}$ = $\frac{20}{35}$ = $\frac{1}{7}$

Hence, the compound probability that if the person surveyed is a smoker then he will be a male = $\frac{1}{7}$
Problem 2:

The probability of raining on Sunday is $0.07$. If today is Sunday then find the probability of rain today.

Solution:


Probability that it is raining and the day is Sunday, $P(A\ and\ B) = 0.07$

Probability that is is Sunday, $P(B)$ = $\frac{1}{7}$

Probability that it will rain if today is Sunday, $P(A|B)$ = $\frac{0.07}{\frac{1}{7}}$ = $0.49$

Hence, the compound probability of raining if it is Sunday is $0.49$.
Problem 3:

In a school the third language has to be chosen between Hindi and French. If a student has taken French then what is the probability that he will take Hindi, if the probability of taking Hindi is 0.34?

Solution:


Probability of taking French and Hindi, $P(A\ and\ B) = 0$ as they are mutually exclusive events.

Probability of taking French, $P(B) = 0.34$

Probability of taking Hindi if French has been opted, $P(A|B)$ = $\frac{P(A and B)}{P(B)}$ = $\frac{0}{0.34}$ = $0$

Compound probability of mutually exclusive events is 0.
Problem 4:

Using a Venn diagram show the probability of occurrence of B if A has already occurred.

Solution:

Here is the venn diagram showing probability of A and B.
Conditional Probability Examples

The shaded area shows the probability of B if A has already occurred.
Conditional Probability Problems
Problem 5:

The given table shows the data of 10 point holders in a given class of 30. Find the probability that the student getting 10 point is a girl.

   10 pointer Not 10 pointer
 Girl   3  8
  Boy   6  13

Solution:

From the table we can retrieve the given information,

Total boys = 19

Total girls = 11

Girl getting 10 point = 3

Probability that the person getting 10 point is a girl = $\frac{3}{11}$

It can be explained using the formula $P(A|B)$ = $\frac{P(A and B)}{P(B)}$ as given here,

Probability of the student being a girl, $P(B)$ = $\frac{11}{30}$

Probability of the student getting 10 point and being a girl, $P(A\ and\ B)$ = $\frac{3}{30}$

Probability that the 10 pointer being selected is  a girl, $P(A|B)$ = $\frac{P(A and B)}{P(B)}$ = $\frac{3}{11}$