Question

The probability of getting a sum of 7, when two dice are thrown simultaneously, is :

• A. \(\frac{1}{9}\)
• B. \(\frac{1}{12}\)
• C. \(\frac{1}{6}\)
• D. \(\frac{5}{36}\)

Hint:

When rolling two dice, the sum of 7 is the most likely outcome.
There is a symmetric pattern for sums of two dice:
- Sums 2 and 12 have 1 way to occur.
- Sums 3 and 11 have 2 ways to occur.
- Sums 4 and 10 have 3 ways to occur.
- Sums 5 and 9 have 4 ways to occur.
- Sums 6 and 8 have 5 ways to occur.
- Sum 7 has 6 ways to occur.
So, the probability of a sum of 7 is always \(\frac{6}{36} = \frac{1}{6}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
Two fair six-sided dice are rolled at the same time. We need to determine the probability that the sum of the numbers showing on top of the two dice is equal to 7.

Step 2: Key Formula or Approach:
1. Each die has 6 outcomes \(\{1, 2, 3, 4, 5, 6\}\).
2. The total number of outcomes when rolling two dice is:
\[ 6 \times 6 = 36 \]
3. Use the basic probability formula:
\[ P(\text{sum of 7}) = \frac{\text{Number of outcomes with a sum of 7}}{\text{Total number of outcomes}} \]

Step 3: Detailed Explanation:
1. Calculate the total number of outcomes:
The sample space contains \(6 \times 6 = 36\) ordered pairs of outcomes of the form \((x, y)\), where \(x\) is the outcome on the first die and \(y\) is the outcome on the second die.
2. List all the outcomes that result in a sum of 7:
Let \(x + y = 7\), where \(1 \le x, y \le 6\). The matching pairs are:
- If \(x = 1 \implies y = 6 \implies (1, 6)\)
- If \(x = 2 \implies y = 5 \implies (2, 5)\)
- If \(x = 3 \implies y = 4 \implies (3, 4)\)
- If \(x = 4 \implies y = 3 \implies (4, 3)\)
- If \(x = 5 \implies y = 2 \implies (5, 2)\)
- If \(x = 6 \implies y = 1 \implies (6, 1)\)
The number of favorable outcomes is 6.
3. Calculate the probability of rolling a sum of 7:
\[ P(\text{sum of 7}) = \frac{6}{36} \]
4. Simplify the fraction:
\[ P(\text{sum of 7}) = \frac{1}{6} \]

Step 4: Final Answer:
The probability of getting a sum of 7 is \(\frac{1}{6}\), which corresponds to option (C).