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🌳 Tree Diagrams & Probability

Grade 9 · Cambridge IGCSE · Probability

Complement

P(A') = 1 − P(A)

Independent Events

P(A and B) = P(A) × P(B)

Mutually Exclusive

P(A or B) = P(A) + P(B)

Tree Diagrams

Multiply along branches, add across

Without Replacement

Second probability changes

Conditional Probability

P(B|A) = P(A∩B) / P(A)

1. Basic Probability Rules

P(event) = number of favourable outcomes / total outcomes
0 ≤ P(A) ≤ 1
Complement: P(A') = 1 − P(A)
If P(rain) = 0.3, then P(no rain) = 1 − 0.3 = 0.7

2. Mutually Exclusive Events

Events are mutually exclusive if they cannot both happen at the same time (e.g., rolling a 1 or a 2 on a die).

P(A or B) = P(A) + P(B)
P(red) = 0.4, P(blue) = 0.3 (cannot both occur).
P(red or blue) = 0.4 + 0.3 = 0.7

3. Independent Events

Events are independent if the outcome of one does not affect the other (e.g., tossing two coins).

P(A and B) = P(A) × P(B)
P(heads) = 0.5, P(six on die) = 1/6 ≈ 0.167. These are independent.
P(heads AND six) = 0.5 × 0.167 ≈ 0.083

4. General Addition Rule

For events that are not mutually exclusive:

P(A or B) = P(A) + P(B) − P(A ∩ B)
P(A) = 0.4, P(B) = 0.3, P(A and B) = 0.1
P(A or B) = 0.4 + 0.3 − 0.1 = 0.6

5. Tree Diagrams

Rules for tree diagrams:
• The probabilities on branches coming from the same point must sum to 1.
• Multiply along branches to get the probability of a combined outcome.
• Add across branches to find the probability of multiple outcomes.
Always check: all final branch probabilities (end nodes) should add up to 1.

6. Without Replacement

When items are selected without replacement, the total number decreases for the second selection, changing the probabilities.

Bag with 3 red and 2 blue balls. Select two without replacement.
P(first red) = 3/5
P(second red | first was red) = 2/4 = 1/2 (only 4 balls left, 2 red)
P(both red) = 3/5 × 2/4 = 6/20 = 3/10 = 0.3

7. Conditional Probability

P(B|A) is read as "the probability of B given A has occurred".

P(B|A) = P(A ∩ B) / P(A)
P(A) = 0.4, P(A ∩ B) = 0.2
P(B|A) = 0.2/0.4 = 0.5
In tree diagrams, conditional probabilities are written on the second set of branches. P(B|A) is the probability on the branch "B given A happened".

Example 1 — Independent Events

P(A) = 0.5, P(B) = 0.4. Find P(A and B) assuming independence.

P(A and B) = P(A) × P(B) = 0.5 × 0.4 = 0.20

Example 2 — Mutually Exclusive

P(A) = 0.3, P(B) = 0.5. A and B are mutually exclusive. Find P(A or B).

P(A or B) = P(A) + P(B) = 0.3 + 0.5 = 0.80

Example 3 — Complement

P(A) = 0.65. Find P(A').

P(A') = 1 − 0.65 = 0.35

Example 4 — Tree Diagram (Without Replacement)

Bag: 4 red, 3 blue. Two drawn without replacement. Find P(both red).

P(1st red) = 4/7
P(2nd red | 1st red) = 3/6 = 1/2
P(both red) = 4/7 × 1/2 = 4/14 = 2/7 ≈ 0.286

Example 5 — Conditional Probability

P(A) = 0.6, P(A and B) = 0.3. Find P(B|A).

P(B|A) = P(A ∩ B) / P(A) = 0.3 / 0.6 = 0.5

Example 6 — Tree Diagram (Check)

P(A) = 0.4. P(B|A) = 0.7, P(B|A') = 0.2. Find P(B).

P(A and B) = 0.4 × 0.7 = 0.28
P(A' and B) = 0.6 × 0.2 = 0.12
P(B) = 0.28 + 0.12 = 0.40

🌳 Tree Diagram Builder

Set the probabilities using sliders. The tree diagram is drawn with all four outcome probabilities auto-calculated.

Adjust sliders and click Draw Tree.

Exercise 1 — P(A and B) for Independent Events

Use P(A and B) = P(A) × P(B). Give answers to 2 d.p.

1. P(A)=0.5, P(B)=0.5. P(A and B)?

2. P(A)=0.3, P(B)=0.2. P(A and B)?

3. P(A)=0.5, P(B)=0.6. P(A and B)?

4. P(A)=0.4, P(B)=0.3. P(A and B)?

5. P(A)=0.3, P(B)=0.5. P(A and B)?

6. P(A)=0.7, P(B)=0.3. P(A and B)?

7. P(A)=0.4, P(B)=0.2. P(A and B)?

8. P(A)=0.7, P(B)=0.5. P(A and B)?

9. P(A)=0.6, P(B)=0.3. P(A and B)?

10. P(A)=0.7, P(B)=0.6. P(A and B)?

Exercise 2 — P(A or B) for Mutually Exclusive Events

1. P(A)=0.3, P(B)=0.4. P(A or B)?

2. P(A)=0.2, P(B)=0.3. P(A or B)?

3. P(A)=0.4, P(B)=0.4. P(A or B)?

4. P(A)=0.1, P(B)=0.5. P(A or B)?

5. P(A)=0.5, P(B)=0.4. P(A or B)?

6. P(A)=0.25, P(B)=0.3. P(A or B)?

7. P(A)=0.45, P(B)=0.3. P(A or B)?

8. P(A)=0.35, P(B)=0.3. P(A or B)?

9. P(A)=0.45, P(B)=0.4. P(A or B)?

10. P(A)=0.2, P(B)=0.25. P(A or B)?

Exercise 3 — Complement P(A') = 1 − P(A)

1. P(A)=0.7. P(A')?

2. P(A)=0.5. P(A')?

3. P(A)=0.8. P(A')?

4. P(A)=0.6. P(A')?

5. P(A)=0.9. P(A')?

6. P(A)=0.55. P(A')?

7. P(A)=0.75. P(A')?

8. P(A)=0.65. P(A')?

9. P(A)=0.85. P(A')?

10. P(A)=0.45. P(A')?

Exercise 4 — Without Replacement (tolerance 0.005)

Give answers as decimals to 3 d.p. The denominators change on the second draw.

1. Bag: 2 red, 4 blue (total 6). P(red then blue) = ?

2. Bag: 2 red, 7 blue (total 9). P(both red) = ?

3. Bag: 5 red, 7 blue (total 12). P(red then red) = ?

4. Bag: 4 red, 11 blue (total 15). P(both red) = ?

5. Bag: 3 red, 7 blue (total 10). P(both red) = ?

6. Bag: 2 red, 9 blue (total 11). P(both red) = ?

7. Bag: 4 red, 6 blue (total 10). P(both red) = ?

8. Bag: 3 red, 9 blue (total 12). P(both red) = ?

9. Bag: 4 red, 6 blue (total 10). P(red then blue) = ?

10. Bag: 3 red, 11 blue (total 14). P(both red) = ?

Exercise 5 — Conditional Probability P(B|A) (tolerance 0.005)

Use P(B|A) = P(A∩B) / P(A). Give decimal answers to 3 d.p.

1. P(A)=0.4, P(A∩B)=0.2. P(B|A)?

2. P(A)=0.6, P(A∩B)=0.2. P(B|A)?

3. P(A)=0.5, P(A∩B)=0.3. P(B|A)?

4. P(A)=0.5, P(A∩B)=0.2. P(B|A)?

5. P(A)=0.4, P(A∩B)=0.1. P(B|A)?

6. P(A)=0.3, P(A∩B)=0.2. P(B|A)?

7. P(A)=0.4, P(A∩B)=0.3. P(B|A)?

8. P(A)=0.5, P(A∩B)=0.1. P(B|A)?

9. P(A)=0.5, P(A∩B)=0.4. P(B|A)?

10. P(A)=0.7, P(A∩B)=0.3. P(B|A)?

🏋️ Practice — 20 Questions

1. P(A)=0.5, P(B)=0.5. P(A and B)?

2. P(A)=0.3, P(B)=0.2. P(A and B)?

3. P(A)=0.5, P(B)=0.6. P(A and B)?

4. P(A)=0.4, P(B)=0.3. P(A and B)?

5. P(A)=0.3, P(B)=0.5. P(A and B)?

6. P(A)=0.7, P(B)=0.3. P(A and B)?

7. P(A)=0.4, P(B)=0.2. P(A and B)?

8. P(A)=0.7, P(B)=0.5. P(A and B)?

9. P(A)=0.6, P(B)=0.3. P(A and B)?

10. P(A)=0.7, P(B)=0.6. P(A and B)?

11. P(A)=0.3, P(B)=0.4 (mut. excl.). P(A or B)?

12. P(A)=0.2, P(B)=0.3 (mut. excl.). P(A or B)?

13. P(A)=0.4, P(B)=0.4. P(A or B)?

14. P(A)=0.1, P(B)=0.5. P(A or B)?

15. P(A)=0.5, P(B)=0.4. P(A or B)?

16. P(A)=0.25, P(B)=0.3. P(A or B)?

17. P(A)=0.45, P(B)=0.3. P(A or B)?

18. P(A)=0.35, P(B)=0.3. P(A or B)?

19. P(A)=0.45, P(B)=0.4. P(A or B)?

20. P(A)=0.2, P(B)=0.25. P(A or B)?

🏆 Challenge — 8 Questions (tolerance 0.005)

1. Bag: 2 red, 4 blue. P(red then blue) without replacement?

2. Bag: 2 red, 7 blue. P(both red) without replacement?

3. Bag: 5 red, 7 blue. P(red then red) without replacement?

4. Bag: 4 red, 11 blue. P(both red) without replacement?

5. Bag: 3 red, 7 blue. P(both red) without replacement?

6. Bag: 2 red, 9 blue. P(both red) without replacement?

7. Bag: 4 red, 6 blue. P(both red) without replacement?

8. Bag: 3 red, 9 blue. P(both red) without replacement?