Conditional probability using a contingency table

Here is how to find the conditional probability using a contingency table that we used in the lesson about marginal probability. The table shows test results for 200 students who took a GED test.

Contingency table

What is conditional probability?

From the list of 200 students, we select a student randomly. However, suppose that you already know the student selected is a male.

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

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

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

Conditional probability

A couple of examples showing how to find the conditional probability using a contingency table

You could compute any of the following 8 conditional probabilities.

  • P(a student has passed /  male)
  • P(a student has passed / female)
  • P(a student has failed / male)
  • P(a student has 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 us compute the P(a student has passed /  male).

If the student is male, then the student will be picked from the list of 102 males.

From this list only 46 students have passed.

Contingency table
P(a student has passed /  male) =  
Number of males who passed / Total number of males

P(a student has passed /  male) =  
46 / 102
= 0.451

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

The number of students who passed is equal to 114.

From this list, only 46 students are males.

P(a student is male / passed) =  
a student is a male / Number of students who passed

P(a student is male / passed) =  
46 / 104
= 0.403

As you can see from the results P(a student has passed /  male) is not equal to

P(a student is male /  passed) because there is a difference.

P(a student has passed /  male): This probability just shows the success rate of males only.

P(a student is male /  passed) : This probability compares the success rate of males to females.

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