Conditional Probability: Definition & Examples

Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

In this lesson you will learn the definition of conditional probability, and how it differs from other kinds of probability. You will also learn how to use the formula for calculating conditional probability and see it being used in two examples. After the lesson, you will take a brief quiz to practice what you learned.

Conditional Situations

When was the last time you asked a friend for a favor, like borrowing their car? They may have responded with, 'you can borrow the car under one condition, you have to take it to the car wash first.' That condition put a restriction on her willingness to let you borrow the car. In other words, you can borrow the car given that you wash it first.

Conditional probability works in a similar way. With conditional probability, we want to find the likelihood of one event occurring given that another event has already occurred. In symbolic form, the probability of event B occurring given event A has occurred would be written

Probability of B given A
Probability of B given A

To calculate the probability of event B occurring given that event A has already occurred, the following formula can be used:

Formula for Conditional Probability
Formula for Conditional Probability

The first fraction would be used when you know the probability of each event occurring. The second fraction would be used when you are given actual values, as you will see in our next example.

Example 1

The following table shows the number of male and female students enrolled in Psychology 101. Find the probability that a student is enrolled in Psychology 101 given that they are female.

Students enrolled in Psychology 101
Students Enrolled in Psychology 101

We might want to say that the probability is 42/235 because there are 42 females that are enrolled in the class out of a total sample space of 235 students. But that isn't the case for this problem.

Notice the word 'given' used in the sentence. That tells us that we are working with conditional probability. Instead of looking at the total 235 students, there is a condition placed on what students we should be focusing on. That condition is that the student is female, so we want to look at the total female students, which is 127. So out of the sample space of 127 students that are female, how many of them are enrolled in Psychology 101? 42 of them. Therefore,

Example 1 Solution
Example 1 Solution

The probability that a student is enrolled in Psychology 101 given that they are female is 33%.

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