How to Draw a Venn Diagram to Calculate Probabilities

There are three common ways to organise data that fall into multiple sets: two-way tables, frequency diagrams and Venn diagrams.  Having blogged about frequency diagrams before I thought I would write about how to draw a Venn Diagram to calculate probabilities.

Recapping Two-Way Tables

How to Draw a Venn Diagram to Calculate Probabilities

This activity works well to review two-way tables from the previous lesson.   I encourage students to use their calculators so little time is wasted on arithmetic.  The probability questions are included to link this with the remainder of the lesson.

How to Draw a Venn Diagram to Calculate Probabilities

The box of the Venn Diagram contains the Universal Set which in this example is the 32 students of the maths class.  Each of the ovals represent the A Level subject, Mathematics and Statistics.  These are called sub-sets. Because a student can choose to study both Mathematics and Statistics the ovals overlap.  This is called the Intersection.  The area contained within the two ovals is called the Union.

We begin by writing the 6 students who choose both subjects in the intersection.  There are 20 students who choose maths and 6 of them also choose statistics.  This means 14 students must be the left most value.  The same method tells you 8 students choose A Level Stats but not Stats and Maths.  There are 4 students outside the Union who do not choose maths or stats at A Level.

After we have calculated the hidden numbers we work through the probability questions.

How to Draw a Venn Diagram to Calculate Probabilities

I ask the class to attempt the next problem in pairs on a single mini-whiteboard.  I find having two students working together on a mini-whiteboard promotes discussion and peer support.  Before they show me their Venn Diagrams I ask the class to think about how to check their working is correct.  Most realise the numbers should add to the total sample of 100.  This helps a couple of pairs rethink and correct their working.

Change of Sample Size

We work through the probability questions one at time.  Problems b) and c) prove simple for most students.  In part d) only half the class realise the sample size has now changed from all the airplanes to only those departing from America.  We discuss the importance of ‘Given that the plane was from America’ in reducing our sample size.

Creating a Three Set Venn Diagram

How to Draw a Venn Diagram to Calculate Probabilities


After 8 minutes all the students have drawn the Venn Diagram confidently and most have found the probability that the student plays piano and drums.  The most able students have also found the probability the student plays the drums given they also play the guitar.

Scheme of Work for Venn Diagrams and Probability

How to draw a Venn Diagram to calculate probabilities is the third lesson in the Probability, Outcomes and Venn Diagrams unit of work.  It follows Calculating Probabilities from Two-Way Tables and precedes Understanding Set Notation.

2 thoughts on “How to Draw a Venn Diagram to Calculate Probabilities

  1. Hollinworth Bryan says:

    Very lucid explanation on both examples, however, for an extension, may incorporate notation.

    For e.g. Union n(A u B) = n(a) + n(b) - n(a).

    Just a suggestion.

    To conclude, excellent resource, thanks for sharing.

    • mrmath_admin says:

      Hi Bryan

      I agree the notation of intersection and union is a good extension to Venn Diagrams.

      I taught this lesson to my middle ability Year 8 class. The follow up in the next lesson was to formalise set notation and to use Venn Diagrams as in introduction to mutually exclusive events.

      Thanks for the feedback

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