Venn diagram questions with answers

Six practice questions with full working: two-set and three-set word problems, “only” and “neither” regions, and probability. Try each one, then reveal the filled-in diagram. Want different numbers? Press New questions.

  1. Question 1 · 2 sets

    29 households were asked which pets they keep.

    • 29 households in total.
    • 16 like a cat.
    • 15 like a dog.
    • 4 like both.
    1. How many like a cat only?
    2. How many like neither?
    3. A person is picked at random. Find P(likes a dog).
    4. Given they like a dog, find the probability they also like a cat.
    UA catA dog

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  2. Question 2 · 2 sets

    20 students were asked which languages they study.

    • 20 students in total.
    • 16 like French.
    • 12 like Spanish.
    • 1 like neither.
    1. How many like both?
    2. How many like exactly one of the two?
    3. Find P(likes French or Spanish or both).
    UFrenchSpanish

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  3. Question 3 · 3 sets

    33 customers were asked which streaming services they pay for.

    • 33 customers in total; 1 chose none of the three.
    • 19 chose service X, 14 chose service Y, 9 chose service Z.
    • 3 chose service X and service Y; 4 chose service X and service Z; 4 chose service Y and service Z.
    • 1 chose all three.
    1. How many chose service X and service Y but not service Z?
    2. How many chose service X only?
    3. How many chose exactly two of the three?
    4. Find P(chose service Z only).
    UService XService YService Z

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  4. Question 4 · 2 sets

    48 people at a café were asked what they drink.

    • 48 people in total.
    • 25 like tea.
    • 25 like coffee.
    • 9 like both.
    1. How many like tea only?
    2. How many like neither?
    3. A person is picked at random. Find P(likes coffee).
    4. Given they like coffee, find the probability they also like tea.
    UTeaCoffee

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  5. Question 5 · 2 sets

    32 people at a café were asked what they drink.

    • 32 people in total.
    • 12 like tea.
    • 17 like coffee.
    • 5 like both.
    1. How many like tea only?
    2. How many like neither?
    3. A person is picked at random. Find P(likes coffee).
    4. Given they like coffee, find the probability they also like tea.
    UTeaCoffee

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  6. Question 6 · 3 sets

    45 pupils were asked which fruits they like.

    • 45 pupils in total; 5 chose none of the three.
    • 15 chose apples, 14 chose bananas, 23 chose cherries.
    • 2 chose apples and bananas; 4 chose apples and cherries; 7 chose bananas and cherries.
    • 1 chose all three.
    1. How many chose apples and bananas but not cherries?
    2. How many chose apples only?
    3. How many chose exactly two of the three?
    4. Find P(chose cherries only).
    UApplesBananasCherries

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Answers

  1. (a) 12   (b) 2   (c) 15/29   (d) 4/15  
  2. (a) 9   (b) 10   (c) 19/20  
  3. (a) 2   (b) 13   (c) 8   (d) 2/33  
  4. (a) 16   (b) 7   (c) 25/48   (d) 9/25  
  5. (a) 7   (b) 8   (c) 17/32   (d) 5/17  
  6. (a) 1   (b) 10   (c) 10   (d) 13/45  

How to solve any Venn diagram word problem

  1. Draw the circles inside a rectangle. The rectangle is everyone in the question, including people in no circle.
  2. Start in the middle. Write the “all of them” number in the centre.
  3. Work outwards. Each overlap of two is the pair total minus the centre. Each “only” part is the set total minus everything already in that circle.
  4. Finish outside. Neither = total − everything inside the circles.
  5. Check. All the regions must add up to the total.

The Venn diagram calculator does these steps for any numbers you type in, and flags facts that contradict each other. The symbols used in exam questions (∪, ∩, A′) are explained on the set notation page.

“And”, “or”, “only”, “neither”

  • A and B (A ∩ B): the overlap.
  • A or B (A ∪ B): anywhere inside either circle, overlap included.
  • Only A: circle A minus every overlap.
  • Neither ((A ∪ B)′): inside the rectangle, outside both circles.
  • Given B: the denominator becomes n(B), not the total.

Need a blank to draw on? Print a two sets with a universal set or three-set template.

Questions people ask

How do you answer Venn diagram questions?

Fill the diagram from the inside out. Put the number in the centre (all sets) first, then each “exactly two” region (pair total minus the centre), then each “only” region (set total minus everything else in that circle), and last the region outside every circle (total minus everything inside). Then read each answer straight off the diagram.

What are common mistakes in Venn diagram questions?

The big one is double counting: writing “12 like both” in the overlap and then also writing the full 30 in the A-only part, instead of 30 − 12 = 18. Others are forgetting the people outside every circle, mixing up “and” (∩, the overlap) with “or” (∪, everything inside either circle), and using the whole total instead of the given set as the denominator in a conditional probability.

How do you find probability from a Venn diagram?

Divide the count of the region you want by the total. For “given B” questions, divide by n(B) instead of the total, because you only look inside circle B.

Are these questions printable?

Yes. Press Print worksheet. Answers print on a separate final section, so you can fold them away or print only the first page.

What are some everyday uses of Venn diagrams?

Comparing two options, sorting survey answers, finding shared items between lists (customers, genes, students), planning compare-and-contrast essays, and solving probability problems.