🎯 Sample Space & Combined Events

Grade 8 · Statistics & Probability · Cambridge Lower Secondary

Sample Space

List of all possible outcomes

Two-Way Tables

Grid for two combined events

Tree Diagrams

Branch paths × probabilities

AND Rule (Independent)

P(A and B) = P(A) × P(B)

OR Rule

P(A or B) = P(A) + P(B) − P(A and B)

1. Sample Space

The sample space is the set of all possible outcomes of an experiment.

Roll a die and flip a coin. The sample space has 6 × 2 = 12 outcomes:
(1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)
Total outcomes = outcomes of event 1 × outcomes of event 2
A sample space diagram (two-way table or list) shows every possible combined outcome systematically. This makes it easy to count favourable outcomes.

2. Two-Way Tables (Sample Space Diagrams)

For two events, draw a grid with one event across the top and the other down the side. Each cell shows a combined outcome.

Two dice — sum of scores:

+123456
1234567
2345678
3456789
45678910
567891011
6789101112

Total = 36 outcomes. P(sum = 7) = 6/36 = 1/6. P(sum > 9) = 6/36 = 1/6.

3. Tree Diagrams

Tree diagrams show combined events step by step. Multiply along branches to find probabilities of combined outcomes.

Flip a coin twice. P(H) = P(T) = 0.5
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
Check: 0.25 × 4 = 1.0 ✓
Multiply along branches to find P(A and B). Add across branches to find P(A or B).

P(at least one head) = P(HH) + P(HT) + P(TH) = 0.25 + 0.25 + 0.25 = 0.75

OR: P(at least one head) = 1 − P(TT) = 1 − 0.25 = 0.75 ✓

4. The AND Rule (Independent Events)

Two events are independent if the outcome of one does not affect the other.

P(A and B) = P(A) × P(B)   [independent events only]
P(rolling 6) = 1/6. P(flipping heads) = 1/2.
P(rolling 6 AND heads) = 1/6 × 1/2 = 1/12
This is equivalent to multiplying along branches of a tree diagram. Each branch must be independent (usually: with replacement, or two different experiments).

5. The OR Rule (General)

For any two events (not necessarily mutually exclusive):

P(A or B) = P(A) + P(B) − P(A and B)

Subtracting P(A and B) avoids counting the overlap twice.

P(even on die) = 3/6. P(> 4 on die) = 2/6. P(even AND > 4) = P(6) = 1/6.
P(even OR > 4) = 3/6 + 2/6 − 1/6 = 4/6 = 2/3
For mutually exclusive events: P(A and B) = 0, so P(A or B) = P(A) + P(B). This is the simpler version you learned in the previous lesson.

Example 1 — Sample Space Diagram

A spinner (1, 2, 3) and a coin (H, T) are used together. List the sample space and find P(odd, heads).

Sample space: (1,H),(1,T),(2,H),(2,T),(3,H),(3,T) → 6 outcomes total
Odd numbers: 1 and 3. P(odd, H) = 2/6 = 1/3
P(even, any) = 2/6 = 1/3 (only 2 is even)

Example 2 — Two Dice Sum Table

Two fair dice are rolled. Find P(sum = 8) and P(sum ≤ 4).

Total outcomes = 36. Sum = 8: (2,6),(3,5),(4,4),(5,3),(6,2) → 5 outcomes
P(sum = 8) = 5/36
Sum ≤ 4: (1,1)→2, (1,2)→3, (2,1)→3, (1,3)→4, (3,1)→4, (2,2)→4 → 6 outcomes
P(sum ≤ 4) = 6/36 = 1/6

Example 3 — Tree Diagram

A bag has 3 red and 2 blue balls. One ball is drawn and replaced, then another is drawn. Find P(both red).

P(R) = 3/5, P(B) = 2/5 (same each draw — replaced)
P(RR) = 3/5 × 3/5 = 9/25
P(RB) = 3/5 × 2/5 = 6/25
P(BR) = 2/5 × 3/5 = 6/25
P(BB) = 2/5 × 2/5 = 4/25
Check: 9+6+6+4 = 25/25 = 1 ✓
P(at least one red) = 1 − P(BB) = 1 − 4/25 = 21/25

Example 4 — OR Rule with Overlap

A card is drawn from 1–10. A = even, B = multiple of 3. Find P(A or B).

Even: {2,4,6,8,10} → P(A) = 5/10
Multiple of 3: {3,6,9} → P(B) = 3/10
Both: {6} → P(A and B) = 1/10
P(A or B) = 5/10 + 3/10 − 1/10 = 7/10

🎲🎲 Two Dice Sample Space

🌳 Tree Diagram Builder

Two independent events:

➕ OR Rule Calculator

Exercise 1 — Sample Space & Counting

1. Spinner (1–4) and coin (H/T). Total outcomes in sample space?

2. Two dice rolled. Total outcomes?

3. Spinner (1–3) and die (1–6). Total outcomes?

4. Two coins flipped. Total outcomes?

5. Spinner (1–4) and die (1–6). P(spinner=2, die=5)?

6. Two coins. P(both heads)?

7. Coin and die. P(heads, even)?

8. Spinner (A,B,C,D) and coin. Total outcomes?

9. Two dice. P(both show 1)?

10. Coin and die. P(tails, < 4)?

Exercise 2 — Two Dice Sums

Two fair dice. Total outcomes = 36.

1. P(sum = 7)?

2. P(sum = 2)?

3. P(sum = 12)?

4. P(sum = 9)?

5. P(sum ≥ 10)?

6. P(sum is even)?

7. P(sum = 5)?

8. P(sum < 5)?

9. P(at least one 6)?

10. P(both dice same)?

Exercise 3 — AND Rule (Independent Events)

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

2. P(heads) = 0.5, P(6) = 1/6. P(heads AND 6)?

3. P(A) = 2/3, P(B) = 1/4. P(A and B)?

4. P(A) = 0.4, P(B) = 0.6. P(A and B)?

5. Die P(even) = 0.5. Flip coin P(H)=0.5. P(even AND H)?

6. P(A) = 1/3, P(B) = 1/3. P(A and B)?

7. P(A) = 0.6, P(B) = 0.7, P(C) = 0.5. P(all three) — independent?

8. P(A) = 0.8, P(B) = 0.9. P(A and B)?

9. P(A) = 3/5, P(B) = 2/3. P(A and B)?

10. P(A) = 0.25, P(B) = 0.4. P(A and B)?

Exercise 4 — Tree Diagrams

A bag: 3 red, 2 blue. Draw with replacement. Give answers as fractions converted to decimals (2 d.p.).

1. P(red, red)?

2. P(blue, blue)?

3. P(red, blue)?

4. P(blue, red)?

5. P(at least one red)?

New bag: P(red) = 0.6, P(blue) = 0.4. Two draws with replacement.

6. P(both red)?

7. P(first red, second blue)?

8. P(exactly one red)?

9. P(both blue)?

10. P(at least one blue)?

Exercise 5 — OR Rule

1. P(A)=0.4, P(B)=0.3, P(A∩B)=0.1. P(A or B)?

2. P(A)=0.5, P(B)=0.4, mutually exclusive. P(A or B)?

3. P(A)=0.6, P(B)=0.5, P(A∩B)=0.3. P(A or B)?

4. P(A)=1/3, P(B)=1/4, mutually exclusive. P(A or B)?

5. P(A or B)=0.8, P(A)=0.5, P(B)=0.4. Find P(A∩B).

6. Die: P(prime)=3/6, P(>4)=2/6, P(prime AND >4)=P(5)=1/6. P(prime or >4)?

7. P(A)=0.7, P(B)=0.3, P(A∩B)=0.15. P(A or B)?

8. P(A)=0.45, P(B)=0.3, P(A∩B)=0.1. P(A or B)?

9. P(A)=2/5, P(B)=3/10, mutually exclusive. P(A or B)?

10. P(A or B)=0.9, P(A∩B)=0.2, P(A)=0.6. Find P(B).

🏋️ Practice — 20 Questions

1. Coin and 4-sided spinner (1-4). Total outcomes?

2. Two dice. P(sum=11)?

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

4. P(A)=0.6, P(B)=0.5, P(A∩B)=0.2. P(A or B)?

5. Two dice. P(sum=6)?

6. Two coins. P(exactly one head)?

7. P(A)=1/4, P(B)=1/5, independent. P(A and B)?

8. Die and coin. P(6 and tails)?

9. P(A)=0.45, P(B)=0.35, ME. P(A or B)?

10. Bag: 4R, 1B. P(R) × P(R) with replacement?

11. Two dice. P(both even)?

12. P(A)=0.8, P(B)=0.7, independent. P(A and B)?

13. P(A)=0.3, P(B)=0.5, P(A∩B)=0.1. P(A or B)?

14. Two dice. P(sum ≤ 3)?

15. P(A)=2/3, P(B)=1/2, independent. P(A and B)?

16. Coin, coin, die. Total outcomes?

17. Two dice. P(sum=10)?

18. P(A)=0.6, P(B)=0.4, ME. P(A or B)?

19. P(A)=0.35, P(B)=0.45, P(A∩B)=0.15. P(A or B)?

20. Two dice. P(difference = 2)?

🏆 Challenge — 8 Questions

1. P(A) = 0.4, P(B) = 0.35, P(A or B) = 0.6. Find P(A and B).

2. A bag has 4 red, 6 blue balls. Two drawn with replacement. P(different colours)?

3. Two dice. P(sum is prime)? Primes between 2 and 12: 2,3,5,7,11.

4. P(A) = 0.7, P(B) = 0.8, independent. P(neither A nor B)?

5. Three independent events each with P = 0.5. P(all three happen)?

6. Die: P(A)=P(even)=0.5, P(B)=P(>3)=0.5, P(A∩B)=P(4 or 6)=2/6. P(A or B)?

7. Two dice. P(sum = 7 OR both dice the same)?

8. P(A and B) = 0.12, P(A) = 0.4, independent. Find P(B).