Tree Diagrams Quizzes

Introduction

Tree diagrams are a visual tool used in GCSE Maths to represent sequential events and calculate probabilities for multi-step experiments. They help students organise outcomes, visualise possible events, and systematically compute probabilities for combined events. Tree diagrams are particularly useful for independent and dependent events and form a key part of probability reasoning.

Core Concepts

What is a Tree Diagram?

A tree diagram is a branching diagram that shows all possible outcomes of one or more experiments in a sequential manner. Each branch represents a possible outcome, and probabilities can be written along the branches to calculate the likelihood of combined events.

Key Terms

• Branch: A line representing a possible outcome of an event.
• Node: A point where branches diverge.
• Independent Events: Events where the outcome of one does not affect the probability of another.
• Dependent Events: Events where the outcome of one affects the probability of another.
• Path: A sequence of branches from the start to an end point, representing a combination of outcomes.
• Multiplication Rule: The probability of a path = product of probabilities along the branches.

Why Use Tree Diagrams?

• Organise multi-step experiments systematically.
• Visualise all possible outcomes.
• Calculate probabilities for combined events accurately.
• Understand sequences and conditional probabilities.

Rules & Steps for Constructing Tree Diagrams

1. Identify the sequence of events.
2. Draw a starting node for the first event.
3. Draw branches for each possible outcome of the first event.
4. From each branch, draw branches for the next event’s possible outcomes.
5. Repeat for all subsequent events.
6. Write the probability for each outcome on the corresponding branch.
7. For each path, multiply the probabilities along the branches to find the probability of that sequence.
8. Sum probabilities of relevant paths to answer “or” questions.

Worked Examples

Example 1: Two Coin Tosses (Independent Events)

Experiment: Toss a coin twice.

• First toss: H or T → probabilities: P(H) = 0.5, P(T) = 0.5
• Second toss: H or T → probabilities: P(H) = 0.5, P(T) = 0.5

Step 1: Draw tree diagram with first toss branches H and T.

Step 2: From each first toss branch, draw branches H and T for the second toss.

Step 3: Calculate probabilities for each path:

• HH: 0.5 × 0.5 = 0.25
• HT: 0.5 × 0.5 = 0.25
• TH: 0.5 × 0.5 = 0.25
• TT: 0.5 × 0.5 = 0.25

Step 4: Event: Exactly one head → paths: HT and TH → P(exactly one head) = 0.25 + 0.25 = 0.5

Example 2: Rolling a Die and Tossing a Coin

Experiment: Roll a six-sided die and toss a coin.

• Die outcomes: 1,2,3,4,5,6 → P(each) = 1/6
• Coin outcomes: H, T → P(each) = 1/2

Step 1: Draw tree diagram with 6 branches for the die.

Step 2: From each die branch, draw 2 branches for coin outcomes.

Step 3: Probability of rolling a 4 and getting heads:

$$ P(4 \text{ and H}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \approx 0.083 $$

Step 4: Probability of rolling an even number and getting tails → sum relevant paths:

• Die even: 2, 4, 6 → each with coin tails → 3 paths
• P(each path) = 1/6 × 1/2 = 1/12
• Total P(even & tails) = 3 × 1/12 = 3/12 = 1/4

Example 3: Dependent Events (Without Replacement)

Experiment: Draw two counters from a bag containing 3 red and 2 blue counters without replacement.

Step 1: First draw probabilities:

• Red: 3/5
• Blue: 2/5

Step 2: Second draw probabilities:

• If first was red: remaining 2 red and 2 blue → P(second red) = 2/4 = 0.5, P(second blue) = 2/4 = 0.5
• If first was blue: remaining 3 red and 1 blue → P(second red) = 3/4 = 0.75, P(second blue) = 1/4 = 0.25

Step 3: Draw tree diagram with first draw branches red/blue, then second draw branches.

Step 4: Calculate probabilities along paths:

• Red → Red: 3/5 × 2/4 = 6/20 = 0.3
• Red → Blue: 3/5 × 2/4 = 6/20 = 0.3
• Blue → Red: 2/5 × 3/4 = 6/20 = 0.3
• Blue → Blue: 2/5 × 1/4 = 2/20 = 0.1

Step 5: Probability of drawing one red and one blue → sum relevant paths = 0.3 + 0.3 + 0.3? careful: paths for exactly one red and one blue = Red→Blue + Blue→Red = 0.3 + 0.3 = 0.6

Common Mistakes

• Forgetting to adjust probabilities for dependent events (without replacement).
• Multiplying probabilities incorrectly along branches.
• Omitting paths or outcomes in the tree diagram.
• Misinterpreting “and” versus “or” when summing probabilities of paths.
• Failing to check that the sum of all path probabilities equals 1.

Applications

Tree diagrams are widely used in exams and real-world contexts:

• Games: Calculating multi-step probabilities in dice, coins, or cards.
• Risk assessment: Sequential events in finance, insurance, or safety planning.
• Quality control: Probability of defects in successive items.
• Genetics: Predicting inheritance of traits using Mendelian ratios.
• Surveys: Sequential or dependent responses analysis.

Strategies & Tips

• Start with the first event and draw branches for each possible outcome.
• Label branches clearly with probabilities.
• For dependent events, adjust probabilities after each branch according to remaining outcomes.
• Multiply probabilities along branches for combined events (“and” rule).
• Sum probabilities of relevant paths for “or” events.
• Check that all possible outcomes are included and that probabilities sum to 1.
• Practice both independent and dependent examples to master tree diagrams.

Summary & Encouragement

Tree diagrams are an essential tool for visualising and calculating probabilities for sequential events. Key points to remember:

• Branches represent possible outcomes; nodes show points of divergence.
• Multiply probabilities along paths to calculate “and” events.
• Sum probabilities of relevant paths to calculate “or” events.
• Adjust probabilities for dependent events carefully.
• Use tree diagrams to organise complex multi-step experiments systematically.

Practice constructing tree diagrams for independent and dependent events, calculating probabilities, and interpreting results. This will improve your confidence and accuracy in GCSE Maths probability. Complete the quizzes to reinforce these skills!