Probability worksheet for calculating the likelihood of sums when rolling two dice.

Let's solve the probability problems step by step based on the sum of two dice.

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Understanding the Problem

When two standard six-sided dice are rolled, there are a total of:

$$
6 \times 6 = 36 \text{ possible outcomes}
$$

The table provided shows all possible sums from rolling two dice. Each cell represents a sum, and since each combination (e.g., die1=1, die2=2) is equally likely, we can count how many outcomes result in each sum or condition.

We will calculate each probability as:

$$
P(\text{event}) = \frac{\text{Number of favorable outcomes}}{36}
$$

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Step-by-Step Solutions

We'll go through each probability one by one.

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#### 1. P(>2) – Probability that the sum is greater than 2

- Total outcomes: 36
- Only sum = 2 is not greater than 2.
- Sum = 2 occurs only once: (1,1)
- So, number of outcomes where sum > 2 = 36 - 1 = 35

$$
P(>2) = \frac{35}{36}
$$

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#### 2. P(≥2) – Probability that the sum is at least 2

- Minimum possible sum is 2, so all outcomes satisfy this.
- All 36 outcomes are ≥ 2

$$
P(≥2) = \frac{36}{36} = 1
$$

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#### 3. P(≤7) – Probability that the sum is less than or equal to 7

Count the number of outcomes with sum ≤ 7.

From the table:

- Sum = 2 → 1 outcome
- Sum = 3 → 2 outcomes
- Sum = 4 → 3 outcomes
- Sum = 5 → 4 outcomes
- Sum = 6 → 5 outcomes
- Sum = 7 → 6 outcomes

Add them up:

$$
1 + 2 + 3 + 4 + 5 + 6 = 21
$$

$$
P(≤7) = \frac{21}{36} = \frac{7}{12}
$$

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#### 4. P(<5) – Probability that the sum is less than 5

Sum < 5 means sum = 2, 3, or 4

- Sum = 2 → 1
- Sum = 3 → 2
- Sum = 4 → 3

Total = 1 + 2 + 3 = 6

$$
P(<5) = \frac{6}{36} = \frac{1}{6}
$$

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#### 5. P(≤5) – Sum ≤ 5

Includes sum = 2, 3, 4, 5

- Sum = 2 → 1
- Sum = 3 → 2
- Sum = 4 → 3
- Sum = 5 → 4

Total = 1 + 2 + 3 + 4 = 10

$$
P(≤5) = \frac{10}{36} = \frac{5}{18}
$$

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#### 6. P(10) – Probability that the sum is exactly 10

From the table, sum = 10 occurs in:
- (4,6), (5,5), (6,4) → 3 outcomes

$$
P(10) = \frac{3}{36} = \frac{1}{12}
$$

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#### 7. P(≥8) – Sum ≥ 8

Sums: 8, 9, 10, 11, 12

Count each:

- Sum = 8 → 5 outcomes
- Sum = 9 → 4 outcomes
- Sum = 10 → 3 outcomes
- Sum = 11 → 2 outcomes
- Sum = 12 → 1 outcome

Total = 5 + 4 + 3 + 2 + 1 = 15

$$
P(≥8) = \frac{15}{36} = \frac{5}{12}
$$

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#### 8. P(<6) – Sum less than 6

Sum = 2, 3, 4, 5

Same as P(≤5), which we already calculated as 10 outcomes.

Wait — but earlier we had P(≤5) = 10 → yes, same here.

$$
P(<6) = \frac{10}{36} = \frac{5}{18}
$$

(Note: <6 means ≤5)

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#### 9. P(≥12) – Sum ≥ 12

Only sum = 12 → 1 outcome

$$
P(≥12) = \frac{1}{36}
$$

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#### 10. P(≤10) – Sum ≤ 10

Total outcomes minus those with sum > 10

Sum > 10 → sum = 11 or 12

- Sum = 11 → 2 outcomes
- Sum = 12 → 1 outcome

So, sum > 10 = 3 outcomes → sum ≤ 10 = 36 - 3 = 33

$$
P(≤10) = \frac{33}{36} = \frac{11}{12}
$$

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#### 11. P(≥6) – Sum ≥ 6

Total outcomes minus sum < 6

Sum < 6 → sum = 2,3,4,5 → we already know this is 10 outcomes

So, sum ≥ 6 = 36 - 10 = 26

$$
P(≥6) = \frac{26}{36} = \frac{13}{18}
$$

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#### 12. P(≤12) – Sum ≤ 12

Maximum sum is 12 → all outcomes satisfy this

$$
P(≤12) = \frac{36}{36} = 1
$$

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#### 13. P(≥11) – Sum ≥ 11

Sum = 11 or 12

- Sum = 11 → 2 outcomes
- Sum = 12 → 1 outcome

Total = 3

$$
P(≥11) = \frac{3}{36} = \frac{1}{12}
$$

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#### 14. P(>5) – Sum > 5

Sum > 5 → sum = 6,7,8,9,10,11,12

We can count:

- Sum = 6 → 5
- Sum = 7 → 6
- Sum = 8 → 5
- Sum = 9 → 4
- Sum = 10 → 3
- Sum = 11 → 2
- Sum = 12 → 1

Total = 5+6+5+4+3+2+1 = 26

Alternatively: total outcomes minus sum ≤ 5 = 36 - 10 = 26

$$
P(>5) = \frac{26}{36} = \frac{13}{18}
$$

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✔ Final Answers:

| Probability | Value |
|-----------|-------|
| P(>2) | $ \frac{35}{36} $ |
| P(≥2) | $ 1 $ |
| P(≤7) | $ \frac{7}{12} $ |
| P(<5) | $ \frac{1}{6} $ |
| P(≤5) | $ \frac{5}{18} $ |
| P(10) | $ \frac{1}{12} $ |
| P(≥8) | $ \frac{5}{12} $ |
| P(<6) | $ \frac{5}{18} $ |
| P(≥12) | $ \frac{1}{36} $ |
| P(≤10) | $ \frac{11}{12} $ |
| P(≥6) | $ \frac{13}{18} $ |
| P(≤12) | $ 1 $ |
| P(≥11) | $ \frac{1}{12} $ |
| P(>5) | $ \frac{13}{18} $ |

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Let me know if you'd like these simplified further or visualized!