GCSE Maths Statistics and Probability - How to Solve: Probability Trees (Conditional & Independent) – Complete Guide

How to Solve: Probability Trees (Conditional & Independent) – Complete Guide

Introduction

"Mastering probability trees lets you crack 6-mark GCSE/A-Level questions on genetics, radioactive decay, and medical trials—questions that can make or break your grade. Examiners love them because they test both maths and science understanding in one go. Get this right, and you’re looking at easy marks."

WHAT YOU NEED TO KNOW FIRST

Before diving into probability trees, you must understand: 1. Basic probability – Calculating P(event) = (number of favourable outcomes) / (total possible outcomes). 2. Independent vs. dependent events – Independent events don’t affect each other (e.g., flipping a coin twice). Dependent events do (e.g., drawing cards without replacement). 3. Multiplication rule for probability – For independent events, P(A and B) = P(A) × P(B).

KEY TERMS & FORMULAS

Key Terms

• Probability tree: A diagram showing all possible outcomes of an event and their probabilities.
• Branch: A line representing one possible outcome.
• Conditional probability: The probability of an event given that another event has already occurred (written as P(A|B)).
• Independent events: Events where the outcome of one does not affect the other.
• Dependent events: Events where the outcome of one affects the other.

Formulas

1. Probability of independent events (AND rule)
2. Formula: P(A and B) = P(A) × P(B)
3. What it means: The probability of both events happening is the product of their individual probabilities.

4. MEMORISE THIS

5. Probability of dependent events (conditional probability)

6. Formula: P(A and B) = P(A) × P(B|A)
7. What it means: The probability of A happening, then B happening given A has already happened.

8. MEMORISE THIS

9. Probability of "at least one" (OR rule)

10. Formula: P(A or B) = P(A) + P(B) – P(A and B)
11. What it means: The probability of either event happening, minus the overlap.
12. Given on exam sheet (but know how to use it!)

STEP-BY-STEP METHOD

Step 1: Identify the events

• Write down the two (or more) events you’re dealing with.
• Are they independent or dependent? This changes how you calculate probabilities.

Step 2: Draw the first branch

• Start with the first event.
• Draw two branches (for binary outcomes, e.g., success/failure, yes/no).
• Label each branch with its probability (e.g., P(success) = 0.6, P(failure) = 0.4).

Step 3: Draw the second branch for each outcome

• From each end of the first branch, draw new branches for the second event.
• If events are independent, the probabilities stay the same.
• If events are dependent, the probabilities change based on the first outcome.

Step 4: Label all probabilities

• Write the probability on every branch.
• For dependent events, use conditional probabilities (e.g., P(B|A)).

Step 5: Calculate combined probabilities

• Multiply along the branches to find the probability of each path.
• Example: P(A and B) = P(A) × P(B|A).

Step 6: Answer the question

• If asked for "both events happening", multiply along the path.
• If asked for "either event happening", add the relevant path probabilities.
• If asked for "at least one", use 1 – P(none).

WORKED EXAMPLES

Example 1 – Basic (Independent Events)

Question: A bag contains 3 red balls and 2 blue balls. A ball is drawn, replaced, and drawn again. What is the probability of drawing two red balls?

Step 1: Identify events - Event 1: First draw. - Event 2: Second draw (with replacement → independent).

Step 2: Draw first branch - P(Red) = 3/5 - P(Blue) = 2/5

Step 3: Draw second branch - Since events are independent, probabilities stay the same. - From "Red" branch: P(Red) = 3/5, P(Blue) = 2/5 - From "Blue" branch: P(Red) = 3/5, P(Blue) = 2/5

Step 4: Label all probabilities - Path 1: Red → Red = (3/5) × (3/5) = 9/25 - Path 2: Red → Blue = (3/5) × (2/5) = 6/25 - Path 3: Blue → Red = (2/5) × (3/5) = 6/25 - Path 4: Blue → Blue = (2/5) × (2/5) = 4/25

Step 5: Calculate combined probability - P(Two reds) = Path 1 = 9/25

What we did and why: - We used the independent events rule because the ball was replaced. - Multiplied along the path to find the probability of both events happening.

Example 2 – Medium (Dependent Events)

Question: A bag contains 3 red balls and 2 blue balls. A ball is drawn and not replaced. What is the probability of drawing two red balls?

Step 1: Identify events - Event 1: First draw. - Event 2: Second draw (without replacement → dependent).

Step 2: Draw first branch - P(Red) = 3/5 - P(Blue) = 2/5

Step 3: Draw second branch - If first draw was Red, now 2 red and 2 blue left → P(Red|First Red) = 2/4 = 1/2 - If first draw was Blue, now 3 red and 1 blue left → P(Red|First Blue) = 3/4

Step 4: Label all probabilities - Path 1: Red → Red = (3/5) × (1/2) = 3/10 - Path 2: Red → Blue = (3/5) × (1/2) = 3/10 - Path 3: Blue → Red = (2/5) × (3/4) = 6/20 = 3/10 - Path 4: Blue → Blue = (2/5) × (1/4) = 2/20 = 1/10

Step 5: Calculate combined probability - P(Two reds) = Path 1 = 3/10

What we did and why: - We used conditional probability because the first draw affected the second. - Adjusted the second probability based on the first outcome.

Example 3 – Exam-Style (Genetics Question)

Question (A-Level Biology): In pea plants, tall (T) is dominant over short (t). A heterozygous tall plant (Tt) is crossed with a short plant (tt). What is the probability that two randomly selected offspring are both tall?

Step 1: Identify events - Event 1: First offspring. - Event 2: Second offspring (independent because one offspring doesn’t affect the other).

Step 2: Determine probabilities from genetics - Parent 1 (Tt) × Parent 2 (tt) → Possible offspring: Tt or tt. - P(Tall) = P(Tt) = 1/2 - P(Short) = P(tt) = 1/2

Step 3: Draw first branch - P(Tall) = 1/2 - P(Short) = 1/2

Step 4: Draw second branch - Independent events → probabilities stay the same. - From "Tall" branch: P(Tall) = 1/2, P(Short) = 1/2 - From "Short" branch: P(Tall) = 1/2, P(Short) = 1/2

Step 5: Label all probabilities - Path 1: Tall → Tall = (1/2) × (1/2) = 1/4 - Path 2: Tall → Short = (1/2) × (1/2) = 1/4 - Path 3: Short → Tall = (1/2) × (1/2) = 1/4 - Path 4: Short → Short = (1/2) × (1/2) = 1/4

Step 6: Answer the question - P(Both tall) = Path 1 = 1/4

What we did and why: - We treated the offspring as independent events (one doesn’t affect the other). - Used the multiplication rule for independent probabilities.

COMMON MISTAKES

1. MISTAKE: Forgetting to adjust probabilities for dependent events.
2. WHY IT HAPPENS: Students assume all events are independent.

3. CORRECT APPROACH: Check if the first event affects the second (e.g., no replacement).

4. MISTAKE: Adding instead of multiplying probabilities.

5. WHY IT HAPPENS: Confusing "AND" (multiply) with "OR" (add).

6. CORRECT APPROACH: "AND" = multiply along branches. "OR" = add path probabilities.

7. MISTAKE: Not labelling all branches.

8. WHY IT HAPPENS: Rushing and missing outcomes.

9. CORRECT APPROACH: Draw every possible path and label probabilities.

10. MISTAKE: Using the wrong denominator for conditional probability.

11. WHY IT HAPPENS: Forgetting the total changes after the first event.

12. CORRECT APPROACH: After first draw, recalculate total possible outcomes.

13. MISTAKE: Misidentifying independent vs. dependent events.

14. WHY IT HAPPENS: Not reading the question carefully (e.g., "with replacement" vs. "without replacement").
15. CORRECT APPROACH: Highlight key words in the question.

EXAM TRAPS

1. TRAP: "At least one" questions.
2. HOW TO SPOT IT: The question asks for "at least one success" or "one or more".

3. HOW TO AVOID IT: Use 1 – P(none) instead of adding all paths.

4. TRAP: Hidden conditional probability.

5. HOW TO SPOT IT: The question gives a probability "given that" something else happened.

6. HOW TO AVOID IT: Look for phrases like "if", "given that", or "assuming".

7. TRAP: Non-binary outcomes (more than two branches).

8. HOW TO SPOT IT: Questions with three or more possible outcomes (e.g., blood types A, B, O).
9. HOW TO AVOID IT: Draw all possible branches, not just two.

1-MINUTE RECAP

"Here’s the fast version for exam day: 1. Identify events – Are they independent or dependent? 2. Draw the tree – First branch for first event, second branches for the next. 3. Label probabilities – Keep them the same for independent events. Adjust for dependent. 4. Multiply along paths – For "AND" questions. 5. Add paths – For "OR" questions. 6. Watch for traps – "At least one" = 1 – P(none). Conditional = adjust probabilities.

If you’re stuck, draw the tree. It forces you to think through every outcome. You’ve got this!"