GCSE Maths Practice: venn-diagrams

Question 5 of 10

This Venn diagram question tests “OR” probability with overlapping groups.

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\( \begin{array}{l}\textbf{A Venn diagram shows information about students who like} \\ \textbf{ice cream and cake in a class of 40 students.} \\ \textbf{25 students like ice cream, 20 like cake, and 10 like both.} \\ \textbf{Find the probability that a randomly chosen student likes} \\ \textbf{ice cream or cake. Give your answer in its simplest form.}\end{array} \)

Diagram

Choose one option:

Fill the overlap first. Then subtract it from each total to get the 'only' regions. Add inside the circles for “or”.

Venn Diagrams and “OR” Probability

Venn diagrams are one of the clearest ways to solve GCSE Maths probability questions involving two groups that overlap. In this question, the two groups are students who like ice cream and students who like cake. The key idea is that some students like both, and that overlap must be handled carefully. If you simply add the totals (25 + 20), you count the students who like both ice cream and cake twice. A Venn diagram prevents this mistake by splitting the information into regions.

How to Fill the Venn Diagram

Draw a rectangle for the whole class (40 students). Inside it, draw two overlapping circles. Label the left circle Ice cream and the right circle Cake.

  1. Fill the overlap first: the question says 10 students like both, so write 10 in the overlapping region.
  2. Find “ice cream only”: 25 students like ice cream in total, but 10 of those are already in the overlap. So ice cream only is \(25 - 10 = 15\). Write 15 in the ice-cream-only region.
  3. Find “cake only”: 20 students like cake in total, but 10 of those are in the overlap. So cake only is \(20 - 10 = 10\). Write 10 in the cake-only region.

Now your diagram is complete for the parts you need. The number who like ice cream or cake is everything inside the circles: \(15 + 10 + 10 = 35\).

Turning the Count into a Probability

Probability means “favourable outcomes divided by total outcomes”. Here, the favourable outcomes are the 35 students who like ice cream or cake. The total is 40 students in the class. So the probability is \(\frac{35}{40}\). Always simplify fractions in GCSE Maths: divide the numerator and denominator by 5 to get \(\frac{7}{8}\).

Worked Examples

Example 1: In a group of 50 people, 28 like tea, 30 like coffee, and 12 like both. Overlap = 12. Tea only = 28 − 12 = 16. Coffee only = 30 − 12 = 18. “Tea or coffee” = 16 + 12 + 18 = 46. Probability = \(\frac{46}{50} = \frac{23}{25}\).

Example 2: In a class of 32 students, 14 do drama, 18 do music, 6 do both. Drama only = 14 − 6 = 8. Music only = 18 − 6 = 12. “Drama or music” = 8 + 6 + 12 = 26. Probability = \(\frac{26}{32} = \frac{13}{16}\).

Common Mistakes

  • Double counting: adding 25 + 20 without subtracting the 10 in the overlap.
  • Putting totals in the regions: writing 25 in the left region instead of splitting into “only” and “both”.
  • Not simplifying: leaving \(\frac{35}{40}\) instead of simplifying to \(\frac{7}{8}\).

Real-Life Use

Overlapping groups appear in real surveys all the time: favourite foods, subjects, sports, streaming services, or clubs. Venn diagrams help businesses and schools understand how interests overlap and avoid counting the same person twice when analysing data.

FAQ

What does “or” mean in Venn diagrams?
It means in one circle, the other circle, or the overlap (both).

Do I need to find “neither” here?
No, because the question asks for “ice cream or cake”. But you can find it to check: students in circles = 35, so neither = 40 − 35 = 5.

What should I do first every time?
Always place the overlap first, then subtract it from each total to find the “only” regions.

Study Tip

If a question mentions two groups and “both”, sketch a Venn diagram immediately. It makes the probability method almost automatic.

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