122i - Two-Way Tables for Combined Events

Lesson 122 — Explicit Instruction: Two-Way Tables for Combined Events

Strand: Probability | Descriptor: AC9M8P02 | Duration: 45 minutes

Learning Intentions

• To construct a two-way table showing all possible combinations of outcomes from two events.
• To use a two-way table to calculate probabilities of combined (joint) outcomes.

Success Criteria

I can:

1. Construct a two-way table for two events, showing every combined outcome.
2. Read joint frequencies and row/column totals correctly from a two-way table.
3. Calculate the probability of a specific combined outcome using table entries.
4. Compare rates within categories to judge whether two variables appear linked.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

1. List all outcomes when a coin is tossed and a die is rolled together.
2. How many outcomes are there in total? How did you get that number without listing every one?
3. A spinner has colours and a coin has sides. How many combined outcomes are possible?
4. Why might a simple list become hard to manage as the number of outcomes grows?

Teacher note: Question 4 is the hook — it motivates the row-and-column structure of a two-way table, introduced in Activity 1.

Activities

Activity 1 — Explicit Instruction: Two-way Tables for Two Independent Events (10 min)

I do: A coin is tossed and a -sector spinner (numbered –) is spun. Build the table:

	1	2	3	4
H	H1	H2	H3	H4
T	T1	T2	T3	T4

Say aloud: “Every cell is one combined outcome. There are equally likely outcomes.” Model: , counting cells H2 and H4.

We do: Together build the table of ordered pairs for two dice (die 1 across the top, die 2 down the side). Use it to find and .

You do: Build the table for a spinner with colours red, blue, green and a coin. Find and .

(Answers: outcomes; ; .)

Activity 2 — Explicit Instruction: Two-way Frequency Tables from Data (10 min)

Real data is often already summarised in a two-way frequency table, rather than listed as individual outcomes.

I do: A survey of students recorded year level and preferred subject:

	Maths	Art	PE	Total
Year 7	18	14	28	60
Year 8	22	16	22	60
Total	40	30	50	120

Model: ”.” ”.” ”.”

We do: Together find , , .

You do: Using the same table, find and (add both column totals, divide by ).

(Answers: ; ; ; ; .)

Activity 3 — Applied Task: Building Your Own Two-way Table (16 min)

Pairs, then share.

A vet surveyed pet owners. owned a dog and owned a cat. Among the dog owners, had desexed their pet. Among the cat owners, had desexed their pet.

1. Organise this information into a two-way table, with row and column totals.
2. Find .
3. Find overall.
4. Within dog owners only, what proportion desexed their pet? Within cat owners only, what proportion desexed their pet?
5. Does desexing status appear to be linked to pet type in this sample? Explain.

Socratic scaffolding for Question 5:

Prompt	Purpose
Understand: what is the question really asking?	Compare the rate of desexing within each pet type, not the raw counts (which differ because there are more dogs than cats).
Devise a plan	Compute the proportion desexed separately for dogs and for cats.
Carry out: dogs
Carry out: cats
Compare	The two rates are identical.
Looking back — what would a different result mean?	If the rates differed, desexing would appear linked to pet type. Because they match exactly, pet type does not seem to affect desexing rate in this sample.

Answer table:

	Desexed	Not desexed	Total
Dog	35	15	50
Cat	21	9	30
Total	56	24	80

Answers: 2. ; 3. ; 4. Dogs: ; Cats: ; 5. The rates are equal ( in each group), so desexing does not appear linked to pet type here.

Checks for Understanding

(6 minutes — exit ticket, collected)

A sample of days recorded whether it rained and whether students were late to school:

	Late	Not late	Total
Rain	8	12	20
No rain	5	25	30
Total	13	37	50

1. How many days had rain?
2. Find .
3. Find .
4. Find the proportion of rainy days that were late, and the proportion of non-rainy days that were late. What does this suggest?

Answers: 1. ; 2. ; 3. ; 4. Rainy days: (); non-rainy days: () — lateness appears roughly times more likely on rainy days, suggesting the two are linked.

Common Misconceptions

Misconception	How to pre-empt it
Reading the wrong cell, or confusing a row total with a column total.	Always label rows and columns clearly, and trace with a finger along the row then down the column before reading a joint cell.
Computing as .	Insist “and” in a two-way table always means one single joint cell, divided by the grand total — never an addition of totals.
Assuming rates must be equal across categories without checking.	Activity 3’s Question 5 and the exit ticket both require an explicit rate comparison, not a guess.
Forgetting to divide by the grand total rather than a row or column subtotal when finding an overall probability.	Model the phrase “out of everyone surveyed” every time an overall probability is calculated.
Confusing a within-group rate (e.g. “of rainy days, how many were late”) with a joint probability (e.g. “rainy and late, out of all days”).	Contrast the two calculations side by side, as in the exit ticket Q3 versus Q4.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). In a class of , students play a sport, play a musical instrument, and do both. Organise this into a two-way table and find how many students do neither.

Answer

	Instrument	No instrument	Total
Sport	5	11	16
No sport	5	7	12
Total	10	18	28

E2 (Kangaroo style). A two-way table has row totals and , and one column totals , of which falls in the first row. Find the missing cell values and the grand total.

Answer

	Column A	Column B	Total
Row 1	15	25	40
Row 2	20	40	60
Total	35	65	100

Row 1, Column B . Row 2, Column A . Row 2, Column B . Grand total .

E3 (Challenge). In a two-way table, the grand total is . One row totals and the other totals . If students were surveyed, find .

Answer

Homework

1. A coin is tossed and a -sided die is rolled. Build the two-way table of all outcomes. Find .
2. A survey of people recorded whether they own a car and whether they own a bike: own both, own a car only, own a bike only, and the rest own neither. Build the two-way table and find .
3. Using the table from Q2, find and .
4. A two-way table shows left-handed and right-handed students. Of the left-handed students, play a sport; of the right-handed students, play a sport. Find the proportion of each group that plays sport, and compare them.
5. Reasoning. Explain why read from a two-way table can never be larger than either or alone.
6. Challenge. A two-way table’s grand total is . The two column totals are in the ratio , and the two row totals are in the ratio . Find both column totals and both row totals.

Answers: 1. outcomes; . 2. Neither ; . 3. ; . 4. Left-handed: ; right-handed: — identical rates, so handedness does not appear linked to sport participation here. 5. The joint cell is always a subset of both the row it sits in and the column it sits in, so its count (and hence its probability) can never exceed either total. 6. Columns: and . Rows: and .