167. Two-Way Tables and Venn Diagrams

Learning Intentions

• Understand that two-way tables and Venn diagrams can be used to show the number of possible outcomes when two different events are considered
• Understand that ‘or’ can mean ‘inclusive or’ or ‘exclusive or’ depending on the context or wording
• Construct a Venn diagram from a worded situation
• Construct a two-way table from a worded situation

Pre-requisite Summary

• Data can be organised into groups based on whether items have certain attributes
• A set is a collection of objects or values
• A Venn diagram uses overlapping circles to show how sets are related
• A two-way table organises data using two different categories at the same time
• The total number in all regions or cells must add to the overall total
• Some items may belong to both sets, one set only, or neither set
• The word “or” can include both possibilities or only one, depending on the wording
• Careful reading of a worded situation is needed before filling in a diagram or table

Worked Examples

Worked Example 1
A class of $20$ students is surveyed.
$12$ like soccer, $9$ like tennis, and $4$ like both soccer and tennis.
Construct a Venn diagram.

Worked Example 2
A group of $30$ students is surveyed.
$18$ have a pet, $14$ play sport, and $7$ do both.
Construct a Venn diagram and find how many do neither.

Worked Example 3
In a survey of $24$ students, $10$ are girls and $14$ are boys.
$6$ girls and $8$ boys wear glasses.
Construct a two-way table.

Worked Example 4
In a survey of $40$ people, $22$ drink tea, $15$ drink coffee, and $6$ drink both.
Explain the difference between:
a) tea or coffee
b) tea or coffee, but not both

Worked Example 5
A class of $28$ students is surveyed.
$16$ study French, $12$ study Music, and $5$ study both.
Construct a Venn diagram and find how many study French or Music.

Worked Example 6
A survey records whether students are in Year $7$ or Year $8$, and whether they travel by bus or not.
There are $12$ Year $7$ students, and $7$ of them travel by bus.
There are $15$ Year $8$ students, and $9$ of them do not travel by bus.
Construct a two-way table.

Problems

Problem 1
A class of $18$ students is surveyed.
$11$ like basketball, $8$ like swimming, and $3$ like both basketball and swimming.
Construct a Venn diagram.

Problem 2
A group of $25$ students is surveyed.
$13$ have a bike, $10$ have a scooter, and $4$ have both.
Construct a Venn diagram and find how many have neither.

Problem 3
In a survey of $20$ students, $9$ are girls and $11$ are boys.
$4$ girls and $6$ boys have a library card.
Construct a two-way table.

Problem 4
In a survey of $36$ people, $20$ like apples, $14$ like bananas, and $5$ like both.
Explain the difference between:
a) apples or bananas
b) apples or bananas, but not both

Problem 5
A class of $26$ students is surveyed.
$15$ play netball, $11$ play volleyball, and $4$ play both.
Construct a Venn diagram and find how many play netball or volleyball.

Problem 6
A survey records whether students are in Year $7$ or Year $8$, and whether they bring lunch from home or not.
There are $13$ Year $7$ students, and $8$ of them bring lunch from home.
There are $14$ Year $8$ students, and $5$ of them do not bring lunch from home.
Construct a two-way table.

Exercises

Understanding and Fluency

1. For each statement, decide whether a Venn diagram or a two-way table would be more suitable.
   a) students who play soccer and/or tennis
   b) students grouped by year level and by whether they wear glasses
   c) people who like tea and coffee

2. In each case, state whether “or” is inclusive or exclusive.
   a) students who study Art or Music
   b) choose soup or salad with your meal
   c) numbers that are even or multiples of $3$

3. A class of $18$ students is surveyed.
   $10$ like dogs, $7$ like cats, and $3$ like both.
   Complete a Venn diagram by finding:
   a) dogs only
   b) cats only
   c) neither

4. A group of $30$ students is surveyed.
   $16$ play football, $12$ play hockey, and $5$ play both.
   Complete a Venn diagram by finding:
   a) football only
   b) hockey only
   c) football or hockey

5. Construct a Venn diagram for this situation.
   In a class of $25$ students, $14$ like reading, $9$ like drawing, and $4$ like both.

6. Construct a Venn diagram for this situation.
   In a survey of $32$ students, $18$ have a pet, $11$ have a sibling, and $6$ have both.
   Also find how many have neither.

7. Construct a two-way table for this situation.
   In a survey of $24$ students, $11$ are girls and $13$ are boys.
   $5$ girls and $7$ boys play an instrument.

8. Construct a two-way table for this situation.
   In a group of $30$ students, $12$ are in Year $7$ and $18$ are in Year $8$.
   $8$ Year $7$ students and $9$ Year $8$ students walk to school.

9. A class survey gives the following information:
   $20$ students in total
   $9$ have glasses
   $11$ do not have glasses
   $8$ are girls
   $4$ girls have glasses
   Construct a two-way table.

10. A survey of $40$ people shows:
    $25$ like pizza
    $18$ like pasta
    $7$ like both
    Find:
    a) pizza only
    b) pasta only
    c) pizza or pasta
    d) pizza or pasta, but not both

Reasoning

11. Explain why the overlap in a Venn diagram represents items that belong to both sets.

12. A student adds $22+15$ to find how many people like tea or coffee, even though $6$ people like both. Explain the error.

13. Explain the difference between inclusive or and exclusive or using your own example.

14. Explain why the row totals and column totals in a two-way table should agree with the overall total.

15. A student places the number for “both” outside the overlap in a Venn diagram. Explain why this is incorrect.

Problem-solving

16. In a class of $35$ students, $19$ play cricket, $13$ play tennis, and $6$ play both.
    a) Construct a Venn diagram
    b) Find how many play neither sport
    c) Find how many play cricket or tennis

17. A survey records whether students are left-handed or right-handed, and whether they wear glasses.
    There are $28$ students.
    $6$ are left-handed and $22$ are right-handed.
    $2$ left-handed students wear glasses and $9$ right-handed students wear glasses.
    Construct a two-way table.

18. A café survey shows that out of $50$ customers, $28$ buy tea, $21$ buy coffee, and $8$ buy both.
    a) Construct a Venn diagram
    b) Find how many buy tea or coffee
    c) Find how many buy neither

19. A club asks members whether they play chess, cards, or both.
    Explain whether the word “or” in “chess or cards” is inclusive or exclusive, and justify your answer in this context.

20. A school survey groups students by whether they are in Year $7$ or Year $8$, and whether they bring a device to school.
    There are $15$ Year $7$ students and $17$ Year $8$ students.
    $9$ Year $7$ students bring a device, and $10$ Year $8$ students do not.
    Construct a two-way table and find the total number of students who bring a device.

Potential Misunderstandings

• Thinking a Venn diagram and a two-way table always show different totals
• Forgetting that the overlap in a Venn diagram represents items in both sets
• Adding both set totals without subtracting the overlap when finding an inclusive “or”
• Confusing inclusive or with exclusive or
• Assuming the word “or” always excludes “both”
• Placing values into a two-way table without checking that row and column totals are consistent
• Forgetting to include the “neither” region in a Venn diagram when there is an overall total
• Treating “both” as belonging to one set only instead of the intersection
• Mixing up row labels and column labels in a two-way table
• Forgetting that all regions or cells must add to the total number in the situation