Combined events: tree diagrams, Venn & two-way tables

When two or more events happen together, you can organise all the possible outcomes using a tree diagram (branching out each possible result step by step), a Venn diagram (overlapping circles showing shared and separate outcomes), or a two-way table (rows and columns cross-referencing two categories). These tools make it far easier to count outcomes and calculate probabilities for combined events than trying to list everything from memory — especially once there's more than one event to track.

Key terms

Tree diagram:

A diagram branching out each possible outcome step by step.

Two-way table:

A table with rows and columns cross-referencing two categories of data.

Questions

1. 1. A tree diagram shows outcomes by:

   • Branching out each possible result step by step
   • Listing everything in one column with no structure
   • Only showing a single outcome
   • Using overlapping circles only

2. 2. A Venn diagram uses:

   • Overlapping circles to show shared and separate outcomes
   • Branching lines only
   • Rows and columns only
   • No visual structure at all

3. 3. A two-way table uses:

   • Rows and columns cross-referencing two categories
   • Only a single row
   • Overlapping circles
   • Branching diagrams only

4. 4. Flipping a coin and rolling a die gives how many total combined outcomes?

   • 12
   • 6
   • 2
   • 8

5. 5. These tools (tree, Venn, two-way table) are used to organise outcomes for:

   • Combined events
   • A single isolated event only
   • Nothing related to probability
   • Only geometry problems

6. 6. The overlapping section of a Venn diagram represents:

   • Outcomes shared by both groups
   • Outcomes in neither group
   • Only one group's outcomes
   • An error in the diagram

7. 7. A two-way table comparing "likes maths" and "likes science" can show:

   • How many students like both, just one, or neither
   • Only how many students exist in total
   • Nothing about student preferences
   • Only the students who like maths

8. 8. A tree diagram for flipping two coins would show how many final branches (outcomes)?

   • 4
   • 2
   • 8
   • 6

9. 9. In a class of 30, a Venn diagram shows 12 play sport, 10 play music, and 5 do both. How many do neither?

   • 13 (30 − (12+10−5))
   • 17
   • 8
   • 5

10. 10. Using a two-way table for 100 students where 40 like maths, 30 like science, and 15 like both: how many like maths only?

    • 25 (40 − 15)
    • 40
    • 15
    • 10

11. 11. Rolling a die twice, how many total outcomes are possible?

    • 36
    • 12
    • 6
    • 2

12. 12. Why might a tree diagram be more useful than a plain list when there are three or more combined events?

    • It organises outcomes systematically, making it easier to count all combinations without missing any
    • Tree diagrams become less useful as more events are added
    • Plain lists are always clearer for three or more events
    • Tree diagrams can only handle a single event

13. 13. From the 12 outcomes of flipping a coin and rolling a die, how many result in "heads and an even number"?

    • 3 (heads with 2, 4, or 6)
    • 6
    • 2
    • 12

14. 14. A Venn diagram with 20 total students, 8 in circle A only, 5 in circle B only, and 4 in the overlap has how many outside both circles?

    • 3 (20 − (8+5+4))
    • 7
    • 17
    • 0

15. 15. Why is a two-way table particularly useful for comparing two categorical variables (like gender and favourite subject)?

    • It clearly cross-references every combination of the two categories in an organised grid
    • It cannot be used to compare two categories at once
    • A two-way table can only ever show one category
    • Two-way tables are less clear than a simple list

16. 16. A survey of 200 people finds 120 like coffee, 90 like tea, and 50 like both. Using this, how many like neither?

    • 40 (200 − (120+90−50))
    • 10
    • 60
    • 200

17. 17. From a full deck of cards and a coin flip, how many total combined outcomes are there (deck has 52 cards)?

    • 104 (52 × 2)
    • 54
    • 52
    • 26

18. 18. Why might organising outcomes with a tree diagram help avoid the common mistake of missing an outcome when counting by hand?

    • Its systematic branching structure ensures every combination is accounted for at each step
    • Tree diagrams are actually more likely to cause missed outcomes than a mental count
    • Missing outcomes is impossible regardless of method used
    • Tree diagrams only work for single, non-combined events

19. 19. A Venn diagram shows 60 people in total: 25 only like hiking, 20 only like camping, and the rest like both. If 5 like neither, how many like both?

    • 10 (60 − 25 − 20 − 5)
    • 15
    • 5
    • 20

20. 20. A market researcher wants to understand overlap between customers who buy Product A, Product B, both, or neither. Why is a Venn diagram especially suited to this question?

    • It visually separates and clearly labels every possible overlap category at once
    • A Venn diagram can only ever show a single product category
    • This kind of overlap cannot be represented visually at all
    • A tree diagram would always be a clearer choice for this specific question

21. 21. Understanding tree diagrams, Venn diagrams and two-way tables mainly helps you to:

    • Systematically organise and count outcomes for combined events to calculate probabilities accurately
    • Avoid ever representing combined events visually
    • Assume all combined events have the same number of outcomes
    • Ignore overlapping outcomes between two groups