Grade 7 Math Chapter 43: Independent and Dependent Events

Chapter Overview

This chapter teaches Independent and Dependent Events with simple language, step-by-step reasoning, and original worked examples. Technical words are explained in plain language.

Key Technical Terms

Independent Events (events whose probabilities do not affect each other)
Dependent Events (events where one outcome changes the probability of another)
Estimate (a close approximation used to check whether an answer is reasonable)
Solution (a value or result that satisfies the problem)
Representation (a way to show mathematics using symbols, diagrams, tables, graphs, or words)
Reasonableness (whether an answer makes sense in the context of the problem)

How to Learn This Chapter

Read the explanation first, follow the examples one step at a time, cover the answer and retry the problem yourself, then use the practice and review questions to check understanding.

43.1 Independent events

Independent events do not affect each other's probabilities. This topic focuses on Independent events .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: Independent events do not affect each other's probabilities. This topic focuses on Independent events .

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: Independent events do not affect each other's probabilities. This topic focuses on Independent events .

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: Independent events do not affect each other's probabilities. This topic focuses on Independent events .

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: Independent events do not affect each other's probabilities. This topic focuses on Independent events .

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: Independent events do not affect each other's probabilities. This topic focuses on Independent events .

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about Independent events. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.2 Dependent events

Dependent events do affect each other's probabilities. This topic focuses on Dependent events .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: Dependent events do affect each other's probabilities. This topic focuses on Dependent events .

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: Dependent events do affect each other's probabilities. This topic focuses on Dependent events .

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: Dependent events do affect each other's probabilities. This topic focuses on Dependent events .

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: Dependent events do affect each other's probabilities. This topic focuses on Dependent events .

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: Dependent events do affect each other's probabilities. This topic focuses on Dependent events .

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about Dependent events. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.3 With replacement

With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on With replacement .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on With replacement .

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on With replacement .

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on With replacement .

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on With replacement .

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on With replacement .

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about With replacement. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.4 Without replacement

With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Without replacement .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Without replacement .

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Without replacement .

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Without replacement .

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Without replacement .

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Without replacement .

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about Without replacement. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.5 Tree diagrams

Tree diagrams is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: Tree diagrams is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: Tree diagrams is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: Tree diagrams is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: Tree diagrams is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: Tree diagrams is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about Tree diagrams. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.6 Organized lists

Organized lists is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Organized lists: Identify a correct example.

State the definition and show why the example fits.
Show all important reasoning.

Very beginner explanation: Organized lists is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: Organized lists: Identify a non-example.

Explain which requirement is missing.
Show all important reasoning.

Very beginner explanation: Organized lists is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: Organized lists: Compare two cases.

State one similarity and one difference.
Show all important reasoning.

Very beginner explanation: Organized lists is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: Organized lists: Apply the idea in a real situation.

Translate the situation into mathematical language.
Show all important reasoning.

Very beginner explanation: Organized lists is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: Organized lists: Create and check your own example.

Use the definition, then verify each condition.
Show all important reasoning.

Very beginner explanation: Organized lists is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain Organized lists in one simple sentence.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Organized lists becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: State the meaning, then give one small example.

Worked Example 7

Problem: Give one example that does not satisfy Organized lists and explain why.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Organized lists becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Identify the rule or condition that fails.

Worked Example 8

Problem: Show Organized lists using a table, diagram, number line, graph, or equation.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Organized lists becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Choose the representation that makes the relationship easiest to see.

Worked Example 9

Problem: After solving a Organized lists problem, how can you check the answer?

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Organized lists becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Use estimation, an inverse operation, substitution, or a second representation.

Worked Example 10

Problem: Give one real-life situation where Organized lists could be useful.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Organized lists becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Connect the idea to money, measurement, data, design, or technology.

Practice Exercise

Create one new question about Organized lists. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.7 Tables of outcomes

Tables of outcomes is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: Tables of outcomes is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: Tables of outcomes is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: Tables of outcomes is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: Tables of outcomes is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: Tables of outcomes is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about Tables of outcomes. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.8 Two-stage experiments

Two-stage experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Two-stage experiments: Identify a correct example.

State the definition and show why the example fits.
Show all important reasoning.

Very beginner explanation: Two-stage experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: Two-stage experiments: Identify a non-example.

Explain which requirement is missing.
Show all important reasoning.

Very beginner explanation: Two-stage experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: Two-stage experiments: Compare two cases.

State one similarity and one difference.
Show all important reasoning.

Very beginner explanation: Two-stage experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: Two-stage experiments: Apply the idea in a real situation.

Translate the situation into mathematical language.
Show all important reasoning.

Very beginner explanation: Two-stage experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: Two-stage experiments: Create and check your own example.

Use the definition, then verify each condition.
Show all important reasoning.

Very beginner explanation: Two-stage experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain Two-stage experiments in one simple sentence.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Two-stage experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: State the meaning, then give one small example.

Worked Example 7

Problem: Give one example that does not satisfy Two-stage experiments and explain why.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Two-stage experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Identify the rule or condition that fails.

Worked Example 8

Problem: Show Two-stage experiments using a table, diagram, number line, graph, or equation.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Two-stage experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Choose the representation that makes the relationship easiest to see.

Worked Example 9

Problem: After solving a Two-stage experiments problem, how can you check the answer?

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Two-stage experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Use estimation, an inverse operation, substitution, or a second representation.

Worked Example 10

Problem: Give one real-life situation where Two-stage experiments could be useful.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Two-stage experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Connect the idea to money, measurement, data, design, or technology.

Practice Exercise

Create one new question about Two-stage experiments. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.9 Multiplication rule introduction

A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication rule introduction .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Find 10% of $80.

Convert 10% to 0.1.
Multiply by 80.

Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication rule introduction .

Answer: $8.00

Worked Example 2

Problem: Find 25% of $60.

Convert 25% to 0.25.
Multiply by 60.

Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication rule introduction .

Answer: $15.00

Worked Example 3

Problem: Find 15% of $120.

Convert 15% to 0.15.
Multiply by 120.

Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication rule introduction .

Answer: $18.00

Worked Example 4

Problem: Find 5% of $250.

Convert 5% to 0.05.
Multiply by 250.

Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication rule introduction .

Answer: $12.50

Worked Example 5

Problem: Find 13% of $75.

Convert 13% to 0.13.
Multiply by 75.

Very beginner explanation: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication rule introduction .

Answer: $9.75

Worked Example 6

Problem: Find 10% of $90.

Convert 10% to 0.1.
Multiply by 90.

Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.

Answer: $9.00

Worked Example 7

Problem: Find 25% of $64.

Convert 25% to 0.25.
Multiply by 64.

Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.

Answer: $16.00

Worked Example 8

Problem: Find 15% of $140.

Convert 15% to 0.15.
Multiply by 140.

Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.

Answer: $21.00

Worked Example 9

Problem: Find 5% of $260.

Convert 5% to 0.05.
Multiply by 260.

Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.

Answer: $13.00

Worked Example 10

Problem: Find 20% of $75.

Convert 20% to 0.2.
Multiply by 75.

Very beginner explanation: Percent of an amount is found by multiplying the amount by the percent written as a decimal.

Answer: $15.00

Practice Exercise

Create one new question about Multiplication rule introduction. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.10 Comparing probabilities with and without replacement

With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Comparing probabilities with and without replacement .

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Comparing probabilities with and without replacement .

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Comparing probabilities with and without replacement .

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Comparing probabilities with and without replacement .

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Comparing probabilities with and without replacement .

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Comparing probabilities with and without replacement .

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about Comparing probabilities with and without replacement. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.11 Marble-bag experiments

Marble-bag experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: Marble-bag experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: Marble-bag experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: Marble-bag experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: Marble-bag experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: Marble-bag experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about Marble-bag experiments. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.12 Card-drawing experiments

Card-drawing experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: Card-drawing experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: Card-drawing experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: Card-drawing experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: Card-drawing experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: Card-drawing experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/4

Worked Example 6

Problem: Explain Card-drawing experiments in one simple sentence.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Card-drawing experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: State the meaning, then give one small example.

Worked Example 7

Problem: Give one example that does not satisfy Card-drawing experiments and explain why.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Card-drawing experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Identify the rule or condition that fails.

Worked Example 8

Problem: Show Card-drawing experiments using a table, diagram, number line, graph, or equation.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Card-drawing experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Choose the representation that makes the relationship easiest to see.

Worked Example 9

Problem: After solving a Card-drawing experiments problem, how can you check the answer?

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Card-drawing experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Use estimation, an inverse operation, substitution, or a second representation.

Worked Example 10

Problem: Give one real-life situation where Card-drawing experiments could be useful.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Card-drawing experiments becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Connect the idea to money, measurement, data, design, or technology.

Practice Exercise

Create one new question about Card-drawing experiments. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.13 Coin-and-die experiments

Coin-and-die experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Roll an even number on a fair die

Favourable outcomes=3.
Total equally likely outcomes=6.
P=3/6.

Very beginner explanation: Coin-and-die experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 3/6

Worked Example 2

Problem: Flip heads on a fair coin

Favourable outcomes=1.
Total equally likely outcomes=2.
P=1/2.

Very beginner explanation: Coin-and-die experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/2

Worked Example 3

Problem: Draw red from 5 red and 3 blue marbles

Favourable outcomes=5.
Total equally likely outcomes=8.
P=5/8.

Very beginner explanation: Coin-and-die experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 5/8

Worked Example 4

Problem: Choose a vowel from A,B,C,E

Favourable outcomes=2.
Total equally likely outcomes=4.
P=2/4.

Very beginner explanation: Coin-and-die experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 2/4

Worked Example 5

Problem: Spin 1 on a 4-equal-section spinner

Favourable outcomes=1.
Total equally likely outcomes=4.
P=1/4.

Very beginner explanation: Coin-and-die experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: 1/4

Worked Example 6

Problem: Find the probability to roll an even number on a fair die.

Favourable outcomes = 3.
Total equally likely outcomes = 6.
Probability = 3/6.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 7

Problem: Find the probability to flip heads on a fair coin.

Favourable outcomes = 1.
Total equally likely outcomes = 2.
Probability = 1/2.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 8

Problem: Find the probability to draw red from 5 red and 3 blue counters.

Favourable outcomes = 5.
Total equally likely outcomes = 8.
Probability = 5/8.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 5/8

Worked Example 9

Problem: Find the probability to choose a vowel from A, B, C, E.

Favourable outcomes = 2.
Total equally likely outcomes = 4.
Probability = 2/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/2

Worked Example 10

Problem: Find the probability to spin section 1 on four equal sections.

Favourable outcomes = 1.
Total equally likely outcomes = 4.
Probability = 1/4.
Simplify if possible.

Very beginner explanation: Probability compares favourable outcomes with all equally likely outcomes.

Answer: 1/4

Practice Exercise

Create one new question about Coin-and-die experiments. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

43.14 Explaining dependence

Explaining dependence is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Beginner Note

Identify what is given, what must be found, choose the correct rule or representation, work one step at a time, and check whether the result makes sense.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: Explaining dependence: Identify a correct example.

State the definition and show why the example fits.
Show all important reasoning.

Very beginner explanation: Explaining dependence is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 2

Problem: Explaining dependence: Identify a non-example.

Explain which requirement is missing.
Show all important reasoning.

Very beginner explanation: Explaining dependence is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 3

Problem: Explaining dependence: Compare two cases.

State one similarity and one difference.
Show all important reasoning.

Very beginner explanation: Explaining dependence is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 4

Problem: Explaining dependence: Apply the idea in a real situation.

Translate the situation into mathematical language.
Show all important reasoning.

Very beginner explanation: Explaining dependence is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 5

Problem: Explaining dependence: Create and check your own example.

Use the definition, then verify each condition.
Show all important reasoning.

Very beginner explanation: Explaining dependence is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Answer: A correct response must satisfy the definition or rule and include a check.

Worked Example 6

Problem: Explain Explaining dependence in one simple sentence.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Explaining dependence becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: State the meaning, then give one small example.

Worked Example 7

Problem: Give one example that does not satisfy Explaining dependence and explain why.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Explaining dependence becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Identify the rule or condition that fails.

Worked Example 8

Problem: Show Explaining dependence using a table, diagram, number line, graph, or equation.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Explaining dependence becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Choose the representation that makes the relationship easiest to see.

Worked Example 9

Problem: After solving a Explaining dependence problem, how can you check the answer?

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Explaining dependence becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Use estimation, an inverse operation, substitution, or a second representation.

Worked Example 10

Problem: Give one real-life situation where Explaining dependence could be useful.

Read the definition of the topic.
Use the definition to build the response.
Check that the response matches every important condition.

Very beginner explanation: Explaining dependence becomes easier to understand when you connect its definition to examples, non-examples, and representations.

Answer: Connect the idea to money, measurement, data, design, or technology.

Practice Exercise

Create one new question about Explaining dependence. Solve it step by step, include units or labels when needed, and check whether your answer is reasonable.

Chapter Notes, Practice, and Common Mistakes

Chapter Notes

Read the complete question before choosing an operation, formula, graph, or model.
Write technical words together with their meaning until you are comfortable using them.
Show all important steps so another student can follow your reasoning.
Keep units, labels, signs, axes, variables, and mathematical symbols clear.
Estimate before or after calculating when an estimate can help check reasonableness.
For real-life problems, explain what the final number means in the situation.

Extra Practice

Create and solve a new question about Independent events . Show your reasoning and check your answer.
Create and solve a new question about Dependent events . Show your reasoning and check your answer.
Create and solve a new question about With replacement . Show your reasoning and check your answer.
Create and solve a new question about Without replacement . Show your reasoning and check your answer.
Create and solve a new question about Tree diagrams . Show your reasoning and check your answer.
Create and solve a new question about Organized lists . Show your reasoning and check your answer.
Create and solve a new question about Tables of outcomes . Show your reasoning and check your answer.
Create and solve a new question about Two-stage experiments . Show your reasoning and check your answer.
Create and solve a new question about Multiplication rule introduction . Show your reasoning and check your answer.
Create and solve a new question about Comparing probabilities with and without replacement . Show your reasoning and check your answer.
Create and solve a new question about Marble-bag experiments . Show your reasoning and check your answer.
Create and solve a new question about Card-drawing experiments . Show your reasoning and check your answer.

Common Mistakes

Using a biased or unrepresentative sample.
Reading graph scales incorrectly.
Treating a misleading graph as trustworthy without checking the axes.
Assuming dependent events are independent.
Using percentages in a circle graph that do not total 100%.

30 Review Questions and Answers

Q1. What is important to remember about Independent events?

Answer: Independent events do not affect each other's probabilities. This topic focuses on Independent events.

Q2. What is important to remember about Dependent events?

Answer: Dependent events do affect each other's probabilities. This topic focuses on Dependent events.

Q3. What is important to remember about With replacement?

Answer: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on With replacement.

Q4. What is important to remember about Without replacement?

Answer: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Without replacement.

Q5. What is important to remember about Tree diagrams?

Answer: Tree diagrams is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q6. What is important to remember about Organized lists?

Answer: Organized lists is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q7. What is important to remember about Tables of outcomes?

Answer: Tables of outcomes is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q8. What is important to remember about Two-stage experiments?

Answer: Two-stage experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q9. What is important to remember about Multiplication rule introduction?

Answer: A tip is an extra amount, often calculated as a percent of a bill. This topic focuses on Multiplication rule introduction.

Q10. What is important to remember about Comparing probabilities with and without replacement?

Answer: With replacement means an item is returned before the next selection, so the original probabilities are restored. This topic focuses on Comparing probabilities with and without replacement.

Q11. What is important to remember about Marble-bag experiments?

Answer: Marble-bag experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q12. What is important to remember about Card-drawing experiments?

Answer: Card-drawing experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q13. What is important to remember about Coin-and-die experiments?

Answer: Coin-and-die experiments is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q14. What is important to remember about Explaining dependence?

Answer: Explaining dependence is an important Grade 7 idea in Independent and Dependent Events. Identify what is known, what must be found, choose an appropriate strategy, show the important steps, and check whether the result is reasonable.

Q15. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q16. Why is estimation useful?

Answer: Estimation helps you judge whether a final answer is reasonable before accepting it.

Q17. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q18. When should you round?

Answer: Usually round near the end of a calculation unless the question specifically asks for earlier rounding.

Q19. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q20. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the relationship connecting them.

Q21. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q22. What should you do if an answer seems unreasonable?

Answer: Re-read the question, check copied values, signs, units, formulas, and calculations.

Q23. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships between lengths, angles, areas, and other quantities.

Q24. Why are tables useful?

Answer: Tables organize values and help reveal patterns, relationships, rates, and missing information.

Q25. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer you obtained.

Q26. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent the same error later.

Q27. How should you study this chapter?

Answer: Review definitions, redo worked examples without looking, practise mixed questions, and explain solutions in your own words.

Q28. When is a calculator useful?

Answer: A calculator is useful for lengthy arithmetic after you understand the mathematical setup and can estimate the expected size of the answer.

Q29. Why compare more than one strategy?

Answer: Different strategies can make a problem easier and provide a way to verify the result.

Q30. What is a mathematical model?

Answer: It is a simplified mathematical representation of a real situation used to analyze, explain, or predict.