Question

Two dice are rolled simultaneously. What is the probability that the sum of the numbers on the two dice is at least 10?

• A. \(\frac{1}{6}\)
• B. \(\frac{1}{9}\)
• C. \(\frac{1}{12}\)
• D. \(\frac{1}{18}\)

Hint:

Tip: List all possible outcomes or use systematic counting to avoid missing any cases.

Answer 1

The Correct Option is A

To find the probability that the sum of the numbers on two dice is at least 10, we first determine all possible outcomes when two dice are rolled. Each die has 6 faces, so there are a total of \(6 \times 6 = 36\) possible outcomes.

Next, we identify the successful outcomes where the sum is at least 10:

• Sum = 10: (4,6), (5,5), (6,4)
• Sum = 11: (5,6), (6,5)
• Sum = 12: (6,6)

There are 6 successful outcomes. The probability is given by the ratio of successful outcomes to total outcomes:

\(\frac{6}{36} = \frac{1}{6}\)

Therefore, the probability that the sum of the numbers on the two dice is at least 10 is \(\frac{1}{6}\).

Answer 2

Step 1: Total possible outcomes 
Since two dice are rolled, total outcomes \( = 6 \times 6 = 36 \).

Step 2: Outcomes with sum \(\geq 10\) 
Possible sums are 10, 11, and 12.

Sum = 10: (4,6), (5,5), (6,4) → 3 outcomes

Sum = 11: (5,6), (6,5) → 2 outcomes

Sum = 12: (6,6) → 1 outcome Total favorable outcomes \(= 3 + 2 + 1 = 6\).

Step 3: Calculate probability 
\[ P = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{6}{36} = \frac{1}{6} \]