Question

What is the probability of getting a sum of \(9\) when two fair dice are rolled simultaneously?

• A. \( \frac{1}{12} \)
• B. \( \frac{1}{9} \)
• C. \( \frac{1}{8} \)
• D. \( \frac{1}{6} \)

Hint:

When two dice are rolled, the total possible outcomes are always \(36\). To find probability, count the number of favorable pairs that produce the required sum.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the probability of a specific event: the sum of the outcomes of rolling two fair six-sided dice is exactly 9.
Step 2: Key Formula or Approach:
The formula for probability is:
\[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \] Step 3: Detailed Explanation:
1. Calculate the Total Number of Possible Outcomes:
Each die has 6 possible outcomes \(\{1, 2, 3, 4, 5, 6\}\).
When two dice are rolled, the total number of combinations is \(6 \times 6 = 36\).
2. Identify the Favorable Outcomes:
We need to find all pairs of outcomes \((d_1, d_2)\) such that \(d_1 + d_2 = 9\).
Listing them out:

If the first die is 3, the second must be 6: \((3, 6)\)
If the first die is 4, the second must be 5: \((4, 5)\)
If the first die is 5, the second must be 4: \((5, 4)\)
If the first die is 6, the second must be 3: \((6, 3)\)
There are a total of 4 favorable outcomes.
3. Calculate the Probability:
Using the probability formula:
\[ P(\text{sum} = 9) = \frac{4}{36} \] Simplifying the fraction:
\[ \frac{4}{36} = \frac{1}{9} \] Step 4: Final Answer:
The probability of getting a sum of 9 is \( \frac{1}{9} \).