Part 1: To find the probability of rolling a sum of 6, 7, or 8, first determine all possible outcomes when rolling two dice (36 outcomes). Then, count the number of outcomes that result in a sum of 6, 7, or 8. Divide this count by the total number of outcomes to get the probability.
Part 2: To find the probability of rolling doubles or a sum of 6 or 8, count the number of outcomes that result in doubles, a sum of 6, or a sum of 8. Use the principle of inclusion-exclusion to avoid double-counting outcomes that satisfy more than one condition. Divide the count by the total number of outcomes.
Part 3: To find the probability of rolling a sum greater than 10, less than 5, or equal to 6, count the number of outcomes that satisfy each condition. Again, use the principle of inclusion-exclusion to avoid double-counting. Divide the count by the total number of outcomes.
To find the probability of rolling a sum of 6, 7, or 8, we first count the outcomes that yield these sums:
- Outcomes for a sum of 6: \( (1,5), (2,4), (3,3), (4,2), (5,1) \) → 5 outcomes
- Outcomes for a sum of 7: \( (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \) → 6 outcomes
- Outcomes for a sum of 8: \( (2,6), (3,5), (4,4), (5,3), (6,2) \) → 5 outcomes
Total outcomes for sums of 6, 7, or 8: \( 5 + 6 + 5 = 16 \).
The probability is given by:
\[
P(6, 7, 8) = \frac{16}{36} = \frac{4}{9}
\]
Next, we calculate the probability of rolling doubles or a sum of 6 or 8:
- Outcomes for doubles: \( (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) \) → 6 outcomes
- Outcomes for a sum of 6: \( (1,5), (2,4), (3,3), (4,2), (5,1) \) → 5 outcomes
- Outcomes for a sum of 8: \( (2,6), (3,5), (4,4), (5,3), (6,2) \) → 5 outcomes
Using the principle of inclusion-exclusion, we find:
- Total outcomes for doubles or a sum of 6 or 8:
\[
6 + 5 + 5 - 1 = 14 \quad (\text{subtracting } (3,3) \text{ which is counted twice})
\]
The probability is:
\[
P(\text{doubles or } 6 \text{ or } 8) = \frac{14}{36} = \frac{7}{18}
\]
Now, we find the probability of rolling a sum greater than 10, less than 5, or equal to 6:
- Outcomes for a sum greater than 10: \( (5,6), (6,5), (6,6) \) → 3 outcomes
- Outcomes for a sum less than 5: \( (1,1), (1,2), (1,3), (2,1), (2,2), (3,1) \) → 6 outcomes
- Outcomes for a sum equal to 6: \( (1,5), (2,4), (3,3), (4,2), (5,1) \) → 5 outcomes
Using the principle of inclusion-exclusion:
- Total outcomes for a sum greater than 10, less than 5, or equal to 6:
\[
3 + 6 + 5 = 14 \quad (\text{no overlaps})
\]
The probability is:
\[
P(>10 \text{ or } <5 \text{ or } =6) = \frac{14}{36} = \frac{7}{18}
\]
- Probability of rolling a sum of 6, 7, or 8: \( \boxed{\frac{4}{9}} \)
- Probability of rolling doubles or a sum of 6 or 8: \( \boxed{\frac{7}{18}} \)
- Probability of rolling a sum greater than 10, less than 5, or equal to 6: \( \boxed{\frac{7}{18}} \)
Answered
