To find the conditional probability, we’ll use the contingency table we used in the lesson on marginal probability. The table shows the test results of 200 students who took a GED test.

Contingency table

We will randomly select a student from the list of 200 students. However, assume that if they are male, you already know the selected student.

The fact that you know the student is male means that the event has already occurred.

Knowing the Student is male, can you calculate that probability that this student has passed or failed.

That kind of probability is called conditional probability and here is the notation to find the probability that a student passed if the student is male, along with some explanations.

Conditional probability

You can calculate one of the following 8 conditional probabilities.

P (a student passed / male)

P (a student passed / female)

P (a student failed / male)

P (a student failed / female)

P (a student is male / passed)

P (a student is male / failed)

P (a student is female / passed)

P (a student is female / failed)

Let’s calculate the P (a student passed / male).

If the student is male, the student will be selected from the list of 102 men.

From this list, there were only 46 students.

P (a student passed / male) =

Number of men who passed
Total number of men

P (a student passed / male) =


= 0.451

What about P (a student is male / passed)?

The number of passed students is 114.

Of that list, only 46 students are male.

P (a student is male / passed) =

A student is a man
Number of passed students

P (a student is male / passed) =


= 0.403

As you can see from the results, P (a student passed / male) is not equal to P (a student is male / passed) because there is a difference.

P (A student passed / male): This probability only shows the success rate of men.

P (A student is male / passed): This probability compares the success rate of men and women.


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