when rolling a two dice the sum is no more than 10 , that is, determine the probability of the event D={2,3,......,10}

To solve this, we need to determine the probability of rolling a sum of no more than 10 with two dice. The event \( D \) represents all sums between 2 and 10 inclusive. Let's break it down step by step.

### Total Possible Outcomes
When rolling two dice, each die has 6 faces, so the total number of possible outcomes is:
\[
6 \times 6 = 36
\]
Thus, there are 36 possible outcomes when rolling two dice.

### Favorable Outcomes
The event \( D \) represents the sums between 2 and 10. Let's count how many outcomes produce these sums:

- **Sum = 2**: (1,1) → 1 outcome
- **Sum = 3**: (1,2), (2,1) → 2 outcomes
- **Sum = 4**: (1,3), (2,2), (3,1) → 3 outcomes
- **Sum = 5**: (1,4), (2,3), (3,2), (4,1) → 4 outcomes
- **Sum = 6**: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 outcomes
- **Sum = 7**: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes
- **Sum = 8**: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 outcomes
- **Sum = 9**: (3,6), (4,5), (5,4), (6,3) → 4 outcomes
- **Sum = 10**: (4,6), (5,5), (6,4) → 3 outcomes

Now, let's add up the number of favorable outcomes:
\[
1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 = 33
\]

### Probability
The probability of the event \( D \) (sum between 2 and 10) is the ratio of favorable outcomes to total outcomes:
\[
P(D) = \frac{33}{36} = \frac{11}{12}
\]

Thus, the probability of the event \( D = \{2, 3, \dots, 10\} \) is \( \frac{11}{12} \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the probability of rolling a sum greater than 10 with two dice?
2. How many outcomes result in a sum of exactly 7 when rolling two dice?
3. What is the probability of rolling a sum less than or equal to 5?
4. What is the probability of rolling a double (e.g., (1,1), (2,2), etc.)?
5. If a third die is added, how would the probability of getting a sum of less than or equal to 10 change?

**Tip:** When calculating probabilities with dice, listing all possible outcomes is a helpful strategy to ensure accuracy.

When rolling two dice, determine the probability of the event D={2,3,......,10} (sum no more than 10).

Find out how to calculate the probability of rolling a sum between 2 and 10 when rolling two dice. This problem demonstrates fundamental probability concepts and counting techniques, suitable for middle school students. Learn how the 33 favorable outcomes lead to a probability of 11/12.

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