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ProbabilityFoundation and Higher (some Higher only)

Tree diagrams, Venn diagrams and tables GCSE questions with answers

8 tree diagrams, venn diagrams and tables questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.

This topic uses tree diagrams, Venn diagrams, two-way tables and frequency trees to organise two events and find probabilities. On Higher you also answer "given that" questions, where you know one thing has already happened.

Quiz yourself: 8 questions

Worked example 1: two-way tables

The two-way table shows how some pupils travel to school. Some of the numbers are missing. How many Year 10 pupils come by bus?

Year 10Year 11TotalWalk1727Bus1222Car191534Total3944
  1. Each row adds up to its total, and each column adds up to its total.
  2. Year 10, Bus: 22 − 12 = 10.

Answer 10

01 / Quiz yourselfTry it, then tap to check

Tree diagrams, Venn diagrams and tables questions to try

Work each one out on paper first. Then open the answer and compare your working with ours, step by step.

01 / Two-way tablesBoth tiers

The two-way table shows how some pupils travel to school. Some of the numbers are missing. How many pupils are there altogether?

Year 10Year 11TotalWalk2433Bus2512Car915Total42
Show the answer and working
  1. Each row adds up to its total, and each column adds up to its total.
  2. Bus total: 25 + 12 = 37.
  3. Overall total: 33 + 37 + 15 = 85.

Answer 85

02 / Frequency treesBoth tiers

61 pupils from Years 7 and 8 went on a school trip. Each pupil chose to visit the museum or the zoo. The frequency tree shows some information about them. How many of the pupils were in Year 8?

Year 7Year 8MuseumZooMuseumZoo6182717
Show the answer and working
  1. The two numbers on the end of each pair of branches add up to the number they come from.
  2. Year 7: 8 + 27 = 35.
  3. Year 8: 61 − 35 = 26.

Answer 26

03 / Tree diagrams: independent eventsBoth tiers

Ali has two spinners, A and B. The probability that spinner A lands on red is 0.5. The probability that spinner B lands on red is 0.4. Ali spins both spinners. The spinners are independent. Some of the probabilities are missing from the tree diagram. Work out the probability that exactly one of the spinners lands on red.

Spinner ASpinner BRed0.5RedNot redNot redRed0.4Not red
Show the answer and working
  1. Each pair of branches adds up to 1. Spinner A: P(Not red) = 1 − 0.5 = 0.5. Spinner B: P(Not red) = 1 − 0.4 = 0.6. The events are independent, so both pairs of Spinner B branches are the same.
  2. There are two routes. Red then Not red: 0.5 × 0.6 = 0.3.
  3. Not red then Red: 0.5 × 0.4 = 0.2.
  4. Add the two routes: 0.3 + 0.2 = 0.5.

Answer 0.5

04 / Venn diagrams: probabilityBoth tiers

The Venn diagram shows information about 56 Year 10 pupils. F = pupils who study French and S = pupils who study Spanish. One of these pupils is chosen at random. Work out the probability that the pupil studies neither French nor Spanish.

ξFS21111212
Show the answer and working
  1. There are 21 + 11 + 12 + 12 = 56 pupils altogether.
  2. Outside both circles: 12.
  3. Probability = 1256, which simplifies to 314.

Answer 314

05 / Completing a Venn diagramBoth tiers

There are 50 Year 10 pupils. 20 study French, 30 study Spanish and 14 study both languages. The Venn diagram shows the 14 who study both languages. How many pupils study only one of the two languages?

ξFS14
Show the answer and working
  1. Start in the overlap: 14 study both languages.
  2. F only: 20 − 14 = 6. S only: 30 − 14 = 16.
  3. Only one: 6 + 16 = 22.

Answer 22

06 / Tree diagrams: without replacementBoth tiers

A bag has 7 yellow counters and 3 red counters. Two counters are taken at random, one after the other, without replacement. Some of the probabilities are missing from the tree diagram. Work out the probability that at least one of the counters is red.

1st counter2nd counterYellow7/10Yellow6/9Red3/9Red3/10YellowRed
Show the answer and working
  1. The missing branches: after a red counter is taken, there are 9 left, 7 yellow and 2 red. So the branches are 79 and 29.
  2. At least one red means every route except Yellow then Yellow.
  3. Yellow then Yellow: 710 × 69 = 4290.
  4. So the probability is 1 − 4290 = 4890.
  5. This simplifies to 815.

Answer 815

07 / Two-way tables: given thatHigher only

Some pupils chose their favourite sport. The two-way table shows the results. Some of the numbers are missing. A girl is chosen at random. Work out the probability that this pupil chose running.

BoysGirlsTotalSwimming2832Tennis2045Running17Total70109
Show the answer and working
  1. Running total: 109 − 32 − 45 = 32.
  2. Girls, Running: 32 − 17 = 15.
  3. Girls total: 109 − 70 = 39.
  4. You only choose from the 39 girls.
  5. 15 of them chose running.
  6. Probability = 1539, which simplifies to 513.

Answer 513

08 / Venn diagrams: set notation and given thatHigher only

The Venn diagram shows information about 41 people at a gym. S = people who swim and W = people who use the weights room. One of the 41 people is chosen at random. Given that the person uses the weights room, work out the probability that the person also swims.

ξSW125213
Show the answer and working
  1. "Given" means you only look at the people in circle W: 21 + 5 = 26.
  2. Of these, 5 are also in S.
  3. Probability = 526.

Answer 526

02 / What's covered8 kinds of question

Tree diagrams, Venn diagrams and tables at GCSE and where marks go

  • Two-way tablesBoth tiers
  • Frequency treesBoth tiers
  • Tree diagrams: independent eventsBoth tiers
  • Venn diagrams: probabilityBoth tiers
  • Completing a Venn diagramBoth tiers
  • Tree diagrams: without replacementBoth tiers
  • Two-way tables: given thatHigher only
  • Venn diagrams: set notation and given thatHigher only

Worked example 2: venn diagrams: set notation and given that

The Venn diagram shows information about 48 people at a gym. S = people who swim and W = people who use the weights room. One of the 48 people is chosen at random. Given that the person uses the weights room, work out the probability that the person also swims.

ξSW212241
  1. "Given" means you only look at the people in circle W: 24 + 2 = 26.
  2. Of these, 2 are also in S.
  3. Probability = 226, which simplifies to 113.

Answer 113

A common mistake

Adding along the branches of a tree diagram. Multiply along the branches for "and", then add the results for "or". If P(rain) = 0.3 each day, P(rain on both days) = 0.3 × 0.3 = 0.09.

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More probability topics

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