Question

Two unbiased dice are thrown simultaneously. Then the possible number of outcomes will be:

• A. 12
• B. 4
• C. 6
• D. 36

Hint:

Remember the general rule for rolling dice:
- 1 Die: \(6^1 = 6\) outcomes
- 2 Dice: \(6^2 = 36\) outcomes
- 3 Dice: \(6^3 = 216\) outcomes
This exponential relationship applies to all independent multi-outcome events.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In probability theory, the sample space represents the set of all possible outcomes of a random experiment.
When multiple independent experiments are conducted together, the total number of outcomes is calculated using the fundamental counting principle.

Step 2: Key Formula or Approach:
If an experiment has \(n\) possible outcomes, and it is repeated \(r\) times (or conducted simultaneously with \(r\) identical independent items), the total number of outcomes is:
\[ \text{Total Outcomes} = n^r \]

Step 3: Detailed Explanation:
Let us apply this formula to the rolling of dice:
1. A single standard fair die has six faces, numbered 1 to 6.
Therefore, rolling one die yields \(n = 6\) possible outcomes:
\[ S_1 = \{1, 2, 3, 4, 5, 6\} \] 2. When two unbiased dice are rolled simultaneously, the outcome of the first die is independent of the second die.
3. According to the multiplication rule, we calculate the total possible outcomes as:
\[ \text{Total Outcomes} = 6 \times 6 = 6^2 = 36 \] 4. These 36 outcomes are represented as ordered pairs \((x, y)\):
\[ S = \{(1,1), (1,2), \dots, (1,6), (2,1), \dots, (6,6)\} \] Therefore, the possible number of outcomes is 36.

Step 4: Final Answer:
The total possible number of outcomes when throwing two dice is 36, matching Option (D).