To calculate the probability of rolling a sum of 7 or 11 with two fair dice, let’s analyze each case separately.

1. Possible sums of 7:

   • There are six pairs of dice that yield a sum of 7:
     • (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
   • This gives a total of 6 outcomes where the sum is 7.

2. Possible sums of 11:

   • There are two pairs of dice that yield a sum of 11:
     • (5, 6) and (6, 5).
   • This gives a total of 2 outcomes where the sum is 11.

3. Total successful outcomes for a sum of 7 or 11:

   • There are $6 + 2 = 8$ successful outcomes that yield a sum of either 7 or 11.

4. Total possible outcomes with two dice:

   • Since each die has 6 faces, there are $6 \times 6 = 36$ total possible outcomes when rolling two dice.

5. Probability calculation:

   • The probability of rolling a sum of 7 or 11 is the ratio of successful outcomes to total outcomes: $\frac{8}{36} = \frac{2}{9}$

So, the probability of rolling a sum of 7 or 11 with two fair dice is $\frac{2}{9}$, or approximately 0.2222 (22.22%).

Would you like more details or have any questions?

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1. What is the probability of rolling a sum of 2 or 12 with two dice?
2. How many ways can you roll a sum of 6 with two dice?
3. What is the probability of rolling an even sum with two dice?
4. How would the probability change if we used three dice to get a sum of 7 or 11?
5. How is the probability affected if the dice are not fair?

Tip: When calculating probabilities with dice, list all combinations to ensure accuracy, especially for sums involving fewer outcomes.