Question

Two dice are rolled simultaneously. What is the probability that the sum of numbers on dice is 9?

• A. 1/9
• B. 1/2
• C. 1/4
• D. 1/6

Hint:

To solve dice-sum problems quickly, memorize this pattern for rolling two dice:
- Sum of 2 or 12 \(\rightarrow\) 1 way
- Sum of 3 or 11 \(\rightarrow\) 2 ways
- Sum of 4 or 10 \(\rightarrow\) 3 ways
- Sum of 5 or 9 \(\rightarrow\) 4 ways (as in this question)
- Sum of 6 or 8 \(\rightarrow\) 5 ways
- Sum of 7 \(\rightarrow\) 6 ways
This symmetrical pattern allows you to find favorable outcomes instantly.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The probability of an event \(E\) occurring is calculated as the ratio of the number of favorable outcomes to the total number of equally likely outcomes in the sample space (\(S\)).

Step 2: Key Formula or Approach:
\[ P(E) = \frac{n(E)}{n(S)} \]

Step 3: Detailed Explanation:
Let us calculate the probability step-by-step:
1. Total Outcomes, \(n(S)\): When two fair, six-sided dice are rolled simultaneously, the total number of possible outcomes in the sample space is:
\[ n(S) = 6 \times 6 = 36 \] 2. Favorable Outcomes, \(n(E)\): We want the sum of the numbers on the two dice to equal 9.
Let the outcome of the first die be \(x\) and the second die be \(y\), such that:
\[ x + y = 9 \] Since the values on the dice must be integers between 1 and 6, the possible pairs \((x, y)\) are:
- If \(x = 3\), then \(y = 6\)
- If \(x = 4\), then \(y = 5\)
- If \(x = 5\), then \(y = 4\)
- If \(x = 6\), then \(y = 3\)
This gives exactly 4 favorable outcomes:
\[ E = \{(3,6), (4,5), (5,4), (6,3)\} \] \[ n(E) = 4 \] 3. Calculate Probability:
\[ P(E) = \frac{n(E)}{n(S)} = \frac{4}{36} = \frac{1}{9} \] Therefore, the probability of rolling a sum of 9 is \(\frac{1}{9}\).

Step 4: Final Answer:
The probability that the sum of the numbers on the dice is 9 is 1/9, matching Option (A).