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
• Draw 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 Use 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 students is surveyed.

like soccer, like tennis, and like both soccer and tennis.

Construct a Venn diagram.

Worked Example 2

A group of students is surveyed.

have a pet, play sport, and do both.

Construct a Venn diagram and Solve how many do neither.

Worked Example 3

In a survey of students, are girls and are boys.

girls and boys wear glasses.

Construct a two-way table.

Worked Example 4

In a survey of people, drink tea, drink coffee, and drink both.

Explain the difference between:

a) tea or coffee

b) tea or coffee, but not both

Worked Example 5

A class of students is surveyed.

study French, study Music, and 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 or Year , and whether they travel by bus or not.

There are Year students, and of them travel by bus.

There are Year students, and of them do not travel by bus.

Construct a two-way table.

Problems

Problem 1

A class of students is surveyed.

like basketball, like swimming, and like both basketball and swimming.

Construct a Venn diagram.

Problem 2

A group of students is surveyed.

have a bike, have a scooter, and have both.

Construct a Venn diagram and find how many have neither.

Problem 3

In a survey of students, are girls and are boys.

girls and boys have a library card.

Construct a two-way table.

Problem 4

In a survey of people, like apples, like bananas, and like both.

Explain the difference between:

a) apples or bananas

b) apples or bananas, but not both

Problem 5

A class of students is surveyed.

play netball, play volleyball, and play both.

Construct a Venn diagram and find how many play netball or volleyball.

Problem 6

A survey records whether students are in Year or Year , and whether they bring lunch from home or not.

There are Year students, and of them bring lunch from home.

There are Year students, and of them do not bring lunch from home.

Construct a two-way table.

Exercises

Understanding and Fluency

Exercise 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

Exercise 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

Exercise 3.

A class of students is surveyed.

like dogs, like cats, and like both.

Complete a Venn diagram by finding:

a) dogs only

b) cats only

c) neither

Exercise 4.

A group of students is surveyed.

play football, play hockey, and play both.

Complete a Venn diagram by finding:

a) football only

b) hockey only

c) football or hockey

Exercise 5.

Construct a Venn diagram for this situation.

In a class of students, like reading, like drawing, and like both.

Exercise 6.

Construct a Venn diagram for this situation.

In a survey of students, have a pet, have a sibling, and have both.

Also find how many have neither.

Exercise 7.

Construct a two-way table for this situation.

In a survey of students, are girls and are boys.

girls and boys play an instrument.

Exercise 8.

Construct a two-way table for this situation.

In a group of students, are in Year and are in Year .

Year students and Year students walk to school.

Exercise 9.

A class survey gives the following information:

students in total

have glasses

do not have glasses

are girls

girls have glasses

Construct a two-way table.

Exercise 10.

A survey of people shows:

like pizza

like pasta

like both

Find:

a) pizza only

b) pasta only

c) pizza or pasta

d) pizza or pasta, but not both

Reasoning

Exercise 11.

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

Exercise 12.

A student adds to find how many people like tea or coffee, even though people like both. Explain the error.

Exercise 13.

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

Exercise 14.

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

Exercise 15.

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

Problem-solving

Exercise 16.

In a class of students, play cricket, play tennis, and play both.

a) Construct a Venn diagram

b) Find how many play neither sport

c) Find how many play cricket or tennis

Exercise 17.

A survey records whether students are left-handed or right-handed, and whether they wear glasses.

There are students.

are left-handed and are right-handed.

left-handed students wear glasses and right-handed students wear glasses.

Construct a two-way table.

Exercise 18.

A café survey shows that out of customers, buy tea, buy coffee, and buy both.

a) Construct a Venn diagram

b) Find how many buy tea or coffee

c) Find how many buy neither

Exercise 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.

Exercise 20.

A school survey groups students by whether they are in Year or Year , and whether they bring a device to school.

There are Year students and Year students.

Year students bring a device, and Year 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

Next: 168r. Reviewing Experimental Probability and Simulation