Using a Two-Way Table to Show Combined Outcomes Revision Notes for Grade 12 NSC Matric Mathematical Literacy

Using a Two-Way Table to Show Combined Outcomes

What is a two-way table?

A two-way table (also called a contingency table) is a powerful tool for organising and displaying the outcomes of two events happening together. Think of it as a grid where one event's outcomes are shown in rows and the other event's outcomes are shown in columns. Each cell in the table represents a possible combined outcome from both events.

Two-way tables work similarly to tree diagrams but present information in a more organised, visual format that makes calculations easier to perform.

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Two-way tables are particularly useful because they provide a clear visual representation of all possible outcomes when two events occur simultaneously, making probability calculations much more straightforward than other methods.

Basic structure of a two-way table

The fundamental structure involves:

Rows: Represent the outcomes of the first event
Columns: Represent the outcomes of the second event
Cells: Show the combined outcomes or frequencies
Totals: Usually included at the edges to show row and column sums

Simple example: Coin tossing

Let's start with a basic example of tossing a coin twice. Each toss can result in either heads (H) or tails (T).

This table shows all possible outcomes when tossing a coin twice:

First coin H, second coin H: (H,H)
First coin H, second coin T: (H,T)
First coin T, second coin H: (T,H)
First coin T, second coin T: (T,T)

Notice there are 4 possible combined outcomes in total.

Worked example 1: Pet ownership survey

Let's examine a real-world example where data has been collected about learners' pets.

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Worked Example: Pet Ownership Survey

Problem: Pumaza surveys 30 learners about their pet ownership:

5 learners have both a cat and a dog
6 learners have a cat but not a dog
12 learners have a dog but not a cat
7 learners have neither a cat nor a dog

Solution: We can organise this information in a two-way table:

	Has a dog	Does not have a dog
Has a cat	5	6
Does not have a cat	12	7

This table clearly shows the relationship between cat ownership and dog ownership among the 30 learners surveyed.

Worked example 2: Rolling two dice

When rolling two dice (a blue die and a red die), we can create a comprehensive table showing all possible outcomes.

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Worked Example: Rolling Two Dice

Setting up the table:

Rows represent the blue die outcomes (B1, B2, B3, B4, B5, B6)
Columns represent the red die outcomes (R1, R2, R3, R4, R5, R6)
Each cell shows the combined outcome, such as (B1; R1)

Key calculations:

Total outcomes: $6 \times 6 = 36$ possible combinations
Probability of any specific outcome: $\frac{1}{36}$
Probability of rolling a 5 on the blue die: $\frac{6}{36} = \frac{1}{6} = 0.17 = 16.67%$

Example probability questions:

Chance of rolling a 3 on blue die (any red): $\frac{6}{36} = \frac{1}{6}$
Chance of rolling (B2; R4): $\frac{1}{36}$
Chance of getting a 1 and a 2 (either die): $\frac{2}{36} = \frac{1}{18}$

Worked example 3: Gift pack probability

Consider a practical scenario involving gift packs for toddlers.

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Worked Example: Gift Pack Probability

The data:

Item	Green	Red	Yellow	Total
Coloured clay	10	15	5	30
Colouring book and crayons	7	20	3	30
Mini chalkboard and chalk	12	25	8	45
Pop-up story book	9	19	5	33
Total	38	79	21	138

Probability calculations:

Probability of selecting a green pack: $\frac{38}{138} = \frac{19}{69}$
Probability of selecting a yellow pack: $\frac{21}{138} = \frac{7}{46}$
Probability of red mini chalkboard: $\frac{25}{138}$
Probability of green coloured clay: $\frac{10}{138} = \frac{5}{69}$

Key steps for probability calculations

When working with two-way tables to calculate probabilities, follow these systematic steps:

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Essential Steps for Two-Way Table Probability Calculations

Identify the favourable outcomes from the table
Count the total number of outcomes (usually the grand total)
Write the probability as a fraction: $P = \frac{\text{favourable outcomes}}{\text{total outcomes}}$
Simplify the fraction where possible
Convert to decimal or percentage if required

Common exam tips

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Critical Exam Success Tips

Always check your totals: Row totals and column totals should add up to the grand total
Read questions carefully: Make sure you understand what specific outcome is being asked for
Simplify fractions: Examiners expect fractions in their lowest terms
Show your working: Even for simple calculations, demonstrate your method
Double-check your counting: It's easy to miscount cells in larger tables

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Key Points to Remember:

A two-way table organises combined outcomes from two events in rows and columns
Each cell represents a possible combination of outcomes from both events
Total outcomes equal the number of filled cells (or grand total for frequency tables)
Probability = favourable outcomes ÷ total outcomes
Always simplify fractions and check your arithmetic carefully

Using a Two-Way Table to Show Combined Outcomes (Grade 12 NSC Matric Mathematical Literacy): Revision Notes