What is the probability that the sum of the three numbers is 11 if 3 dices are thrown simultaneously?

To calculate the probability that the sum of the numbers on three dice equals 11 when thrown simultaneously, we need to determine the total number of possible outcomes and the number of favorable outcomes where the sum is exactly 11.

### Step 1: Total number of outcomes
Each die has 6 faces (numbered 1 to 6), so for 3 dice, the total number of possible outcomes is:
\[
6 \times 6 \times 6 = 216
\]

### Step 2: Number of favorable outcomes
We need to find all the combinations of the numbers on the three dice that add up to 11. We are looking for triples \((x_1, x_2, x_3)\), where each \(x_i \in \{1, 2, 3, 4, 5, 6\}\) and \(x_1 + x_2 + x_3 = 11\).

We can enumerate all the valid combinations of dice rolls that sum to 11:

- (5, 5, 1)
- (5, 4, 2)
- (5, 3, 3)
- (4, 4, 3)
- (4, 5, 2)
- (6, 3, 2)
- (6, 4, 1)

These are just some combinations, but we can have different orders of these numbers since dice rolls are independent.

After calculating all possible ordered combinations, we find there are **27 favorable outcomes**.

### Step 3: Calculate the probability
The probability is given by the ratio of favorable outcomes to the total number of possible outcomes:
\[
P(\text{sum} = 11) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{27}{216} = \frac{1}{8}
\]

Thus, the probability that the sum of the three dice is 11 is:
\[
\boxed{\frac{1}{8}}
\]

Would you like further details on any step or the enumeration of all combinations?

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Here are five related questions to expand on this concept:

1. What is the probability of getting a sum of 10 when rolling three dice?
2. How would you calculate the probability of getting a sum of 12 with three dice?
3. If you roll two dice, what is the probability that the sum is 7?
4. How does the probability distribution of sums change as more dice are rolled?
5. What is the probability of getting the same number on all three dice?

**Tip**: When working with dice, it's helpful to remember that each die is independent, meaning the outcome of one die doesn’t affect the others.

Discover how to calculate the probability of getting a sum of 11 when rolling three dice simultaneously. This problem covers essential probability concepts, including total and favorable outcomes, with a step-by-step guide to solving it. Suitable for students in Grades 6-8.

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