Sample Space Diagrams Worksheet | Cazoom Maths Worksheets - Free Printable

Let's solve each section of this Sample Space Diagrams worksheet step by step.

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✔ Section A: Dice Probability

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## 🔹 Question 1: Two fair dice are thrown and the scores are ADDED together.

We are adding two dice (each with faces 1 to 6).

Step 1: Complete the sample space diagram

We create a 6×6 grid where:
- Rows = First die
- Columns = Second die
- Each cell = Sum of the two dice

Here’s the completed table:

| | 1 | 2 | 3 | 4 | 5 | 6 |
|-------|-----|-----|-----|-----|-----|-----|
| 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| 6 | 7 | 8 | 9 | 10 | 11 | 12 |

✔ Total outcomes:
There are $6 \times 6 = 36$ possible outcomes.

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Now answer the questions:

#### ➤ What is the probability of getting a score of 5?

Find how many times 5 appears in the table:
- (1,4), (2,3), (3,2), (4,1) → 4 times

$$
P(\text{score of } 5) = \frac{4}{36} = \frac{1}{9}
$$

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#### ➤ What is the probability of getting a score greater than 9?

Scores > 9: 10, 11, 12

Count them:
- 10: (4,6), (5,5), (6,4) → 3
- 11: (5,6), (6,5) → 2
- 12: (6,6) → 1
→ Total = 3 + 2 + 1 = 6

$$
P(\text{> 9}) = \frac{6}{36} = \frac{1}{6}
$$

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#### ➤ What is the probability of getting a score less than 7?

Scores < 7: 2, 3, 4, 5, 6

Count them:
- 2: (1,1) → 1
- 3: (1,2), (2,1) → 2
- 4: (1,3), (2,2), (3,1) → 3
- 5: (1,4), (2,3), (3,2), (4,1) → 4
- 6: (1,5), (2,4), (3,3), (4,2), (5,1) → 5
→ Total = 1+2+3+4+5 = 15

$$
P(\text{< 7}) = \frac{15}{36} = \frac{5}{12}
$$

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## 🔹 Question 2: Two fair dice are thrown and the scores are MULTIPLIED together.

Now we multiply the two dice.

Step 1: Complete the sample space diagram

| | 1 | 2 | 3 | 4 | 5 | 6 |
|-------|-----|-----|-----|-----|-----|-----|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 |

✔ Total outcomes: Still $6 \times 6 = 36$

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Answer the questions:

#### ➤ What is the probability of getting a score of 6?

Find all pairs that multiply to 6:
- (1,6), (2,3), (3,2), (6,1) → 4 times

$$
P(\text{score of } 6) = \frac{4}{36} = \frac{1}{9}
$$

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#### ➤ What is the probability of getting a score greater than 16?

Look for products > 16:
- 18: (3,6), (6,3)
- 20: (4,5), (5,4)
- 24: (4,6), (6,4)
- 30: (5,6), (6,5)
- 36: (6,6)

List:
- 18 → 2
- 20 → 2
- 24 → 2
- 30 → 2
- 36 → 1
→ Total = 2+2+2+2+1 = 9

$$
P(\text{> 16}) = \frac{9}{36} = \frac{1}{4}
$$

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#### ➤ What is the probability of getting a score less than 4?

Products < 4: 1, 2, 3

- 1: (1,1) → 1
- 2: (1,2), (2,1) → 2
- 3: (1,3), (3,1) → 2
→ Total = 1+2+2 = 5

$$
P(\text{< 4}) = \frac{5}{36}
$$

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## 🔹 Question 3: A six-faced dice and a four-faced dice are thrown. The first score is SUBTRACTED FROM the second score.

So: Second score – First score

Note:
- Six-faced die: 1–6
- Four-faced die: 1–4

So rows = second die (1–4), columns = first die (1–6)

Step 1: Complete the sample space diagram

| | 1 | 2 | 3 | 4 | 5 | 6 |
|-------|-----|-----|-----|-----|-----|-----|
| 1 | 0 | -1 | -2 | -3 | -4 | -5 |
| 2 | 1 | 0 | -1 | -2 | -3 | -4 |
| 3 | 2 | 1 | 0 | -1 | -2 | -3 |
| 4 | 3 | 2 | 1 | 0 | -1 | -2 |

✔ Total outcomes: $6 \times 4 = 24$

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Answer the questions:

#### ➤ What is the probability of getting a negative score?

Negative means result < 0.

Count cells with negative values:

From table:
- Row 1 (second=1): -1, -2, -3, -4, -5 → 5 negatives
- Row 2 (second=2): -1, -2, -3, -4 → 4
- Row 3 (second=3): -1, -2, -3 → 3
- Row 4 (second=4): -1, -2 → 2
→ Total = 5+4+3+2 = 14

$$
P(\text{negative}) = \frac{14}{24} = \frac{7}{12}
$$

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#### ➤ What is the probability of getting a score ≥ 0?

This includes 0 and positive numbers.

Total outcomes = 24
Negative outcomes = 14
So non-negative = 24 – 14 = 10

Alternatively, count:
- 0: appears in (1,1), (2,2), (3,3), (4,4) → 4
- Positive: 1,2,3
- 1: (2,1), (3,2), (4,3) → 3
- 2: (3,1), (4,2) → 2
- 3: (4,1) → 1
→ Total = 4 + 3 + 2 + 1 = 10

$$
P(\geq 0) = \frac{10}{24} = \frac{5}{12}
$$

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## ✔ Final Answers Summary

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Question 1: Addition

- Sample space completed.
- Total outcomes: 36
- P(score of 5) = $\frac{1}{9}$
- P(score > 9) = $\frac{1}{6}$
- P(score < 7) = $\frac{5}{12}$

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Question 2: Multiplication

- Sample space completed.
- Total outcomes: 36
- P(score of 6) = $\frac{1}{9}$
- P(score > 16) = $\frac{1}{4}$
- P(score < 4) = $\frac{5}{36}$

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Question 3: Subtraction (Second – First)

- Sample space completed.
- Total outcomes: 24
- P(negative score) = $\frac{7}{12}$
- P(score ≥ 0) = $\frac{5}{12}$

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