Question

Two dice are thrown together. The probability of getting a sum more than 8 is

• A. \( \frac{7}{36} \)
• B. \( \frac{5}{12} \)
• C. \( \frac{5}{18} \)
• D. \( \frac{7}{18} \)

Hint:

To find the probability of a specific event, count the favorable outcomes and divide by the total number of possible outcomes.

Answer 1

The Correct Option is C

Step 1: Total possible outcomes.
When two dice are thrown, the total number of outcomes is \( 6 \times 6 = 36 \).
Step 2: Favorable outcomes.
The sums greater than 8 are: 9, 10, 11, and 12. We list the pairs that produce these sums: - Sum 9: (3, 6), (4, 5), (5, 4), (6, 3) — 4 outcomes - Sum 10: (4, 6), (5, 5), (6, 4) — 3 outcomes - Sum 11: (5, 6), (6, 5) — 2 outcomes - Sum 12: (6, 6) — 1 outcome Thus, the total number of favorable outcomes is \( 4 + 3 + 2 + 1 = 10 \).
Step 3: Probability calculation.
The probability of getting a sum greater than 8 is: \[ P = \frac{10}{36} = \frac{5}{18} \]
Step 4: Conclusion.
Therefore, the correct answer is (3) \( \frac{5}{18} \).

Answer 2

Instead of directly counting the outcomes where the sum exceeds 8, it's often quicker to count the complementary event, sums of 8 or less, and subtract from 1, since \( 6\times6=36 \) is a small, fixed sample space either way.

Sums from 2 to 8 arise from the following pair counts: sum 2 (1 way), sum 3 (2), sum 4 (3), sum 5 (4), sum 6 (5), sum 7 (6), sum 8 (5). Adding these: \( 1+2+3+4+5+6+5=26 \). So \( P(\text{sum}\le 8)=\frac{26}{36}=\frac{13}{18} \), and therefore \( P(\text{sum}>8)=1-\frac{13}{18}=\frac{5}{18} \).

1. \( \frac{7}{36} \): This is smaller than our computed \( \frac{10}{36} \) favorable count (since \( \frac{5}{18}=\frac{10}{36} \)), so it undercounts the favorable outcomes.
2. \( \frac{5}{12} \): This equals \( \frac{15}{36} \), larger than the true \( \frac{10}{36} \), overcounting the favorable outcomes.
3. \( \frac{5}{18} \): This equals \( \frac{10}{36} \), matching exactly \( 1-\frac{26}{36} \) derived from the complement.
4. \( \frac{7}{18} \): This equals \( \frac{14}{36} \), again larger than the correct \( \frac{10}{36} \).

The complement calculation confirms the favorable count is 10 out of 36 total outcomes.

Therefore, the correct answer is \( \frac{5}{18} \).