This worksheet helps students visualize probability outcomes by completing sample space diagrams for dice games involving addition, multiplication, and subtraction.

Problem Analysis and Solution

The worksheet involves creating sample space diagrams for different scenarios involving dice rolls and calculating probabilities based on those diagrams. Let's solve each part step by step.

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#### Section A: Dice Probability

##### 1) Two fair dice are thrown, and the scores are ADDED together.

- Step 1: Complete the sample space diagram.
- Each die has 6 faces, so there are \(6 \times 6 = 36\) possible outcomes.
- The sample space diagram will have rows and columns representing the outcomes of the two dice. The entries in the table will be the sums of the numbers on the two dice.

| | 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 |

- Step 2: How many outcomes are there?
- There are \(6 \times 6 = 36\) outcomes.

- Step 3: What is the probability of getting a score of '5'?
- From the table, the pairs that sum to 5 are: (1, 4), (2, 3), (3, 2), (4, 1).
- There are 4 favorable outcomes.
- Probability = \(\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{4}{36} = \frac{1}{9}\).

- Step 4: What is the probability of getting a score greater than '9'?
- From the table, the pairs that sum to more than 9 are: (4, 6), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6).
- There are 6 favorable outcomes.
- Probability = \(\frac{6}{36} = \frac{1}{6}\).

- Step 5: What is the probability of getting a score less than '7'?
- From the table, the pairs that sum to less than 7 are: (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (4, 1), (4, 2), (5, 1).
- There are 15 favorable outcomes.
- Probability = \(\frac{15}{36} = \frac{5}{12}\).

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##### 2) Two fair dice are thrown, and the scores are MULTIPLIED together.

- Step 1: Complete the sample space diagram.
- Each die has 6 faces, so there are \(6 \times 6 = 36\) possible outcomes.
- The sample space diagram will have rows and columns representing the outcomes of the two dice. The entries in the table will be the products of the numbers on the two dice.

| | 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 |

- Step 2: How many outcomes are there?
- There are \(6 \times 6 = 36\) outcomes.

- Step 3: What is the probability of getting a score of '6'?
- From the table, the pairs that multiply to 6 are: (1, 6), (2, 3), (3, 2), (6, 1).
- There are 4 favorable outcomes.
- Probability = \(\frac{4}{36} = \frac{1}{9}\).

- Step 4: What is the probability of getting a score greater than '16'?
- From the table, the pairs that multiply to more than 16 are: (4, 5), (5, 4), (4, 6), (6, 4), (5, 5), (5, 6), (6, 5), (6, 6).
- There are 8 favorable outcomes.
- Probability = \(\frac{8}{36} = \frac{2}{9}\).

- Step 5: What is the probability of getting a score less than '4'?
- From the table, the pairs that multiply to less than 4 are: (1, 1), (1, 2), (1, 3), (2, 1), (3, 1).
- There are 5 favorable outcomes.
- Probability = \(\frac{5}{36}\).

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

- Step 1: Complete the sample space diagram.
- The six-faced die has 6 faces, and the four-faced die has 4 faces, so there are \(6 \times 4 = 24\) possible outcomes.
- The sample space diagram will have rows and columns representing the outcomes of the two dice. The entries in the table will be the result of subtracting the first score from the second score.

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

- Step 2: How many outcomes are there?
- There are \(6 \times 4 = 24\) outcomes.

- Step 3: What is the probability of getting a negative score?
- From the table, the negative scores are: -1, -2, -3, -4, -5.
- Counting all occurrences of negative scores:
- -1 appears 5 times.
- -2 appears 4 times.
- -3 appears 3 times.
- -4 appears 2 times.
- -5 appears 1 time.
- Total negative outcomes = \(5 + 4 + 3 + 2 + 1 = 15\).
- Probability = \(\frac{15}{24} = \frac{5}{8}\).

- Step 4: What is the probability of getting a score ≥ 0?
- From the table, the non-negative scores are: 0, 1, 2, 3.
- Counting all occurrences of non-negative scores:
- 0 appears 4 times.
- 1 appears 3 times.
- 2 appears 2 times.
- 3 appears 1 time.
- Total non-negative outcomes = \(4 + 3 + 2 + 1 = 10\).
- Probability = \(\frac{10}{24} = \frac{5}{12}\).

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Final Answers:

1.
- Number of outcomes: \(36\)
- Probability of getting a score of '5': \(\boxed{\frac{1}{9}}\)
- Probability of getting a score greater than '9': \(\boxed{\frac{1}{6}}\)
- Probability of getting a score less than '7': \(\boxed{\frac{5}{12}}\)

2.
- Number of outcomes: \(36\)
- Probability of getting a score of '6': \(\boxed{\frac{1}{9}}\)
- Probability of getting a score greater than '16': \(\boxed{\frac{2}{9}}\)
- Probability of getting a score less than '4': \(\boxed{\frac{5}{36}}\)

3.
- Number of outcomes: \(24\)
- Probability of getting a negative score: \(\boxed{\frac{5}{8}}\)
- Probability of getting a score ≥ 0: \(\boxed{\frac{5}{12}}\)