Describe the sample space of this experiment :
Two dice are rolled .

Answer

To describe the sample space for the experiment of rolling two dice, we can follow these steps:

### Step 1: Understand the Experiment
The experiment involves rolling two six-sided dice. Each die has faces numbered from 1 to 6.

### Step 2: Define the Sample Space
The sample space is the set of all possible outcomes of the experiment. Since we are rolling two dice, we can represent the outcome as an ordered pair (d1, d2), where d1 is the result from the first die and d2 is the result from the second die.

### Step 3: List Possible Outcomes
For each die, the possible outcomes are 1, 2, 3, 4, 5, and 6. Therefore, we can list the outcomes systematically:

- If the first die shows 1, the second die can show:
- (1, 1)
- (1, 2)
- (1, 3)
- (1, 4)
- (1, 5)
- (1, 6)

- If the first die shows 2, the second die can show:
- (2, 1)
- (2, 2)
- (2, 3)
- (2, 4)
- (2, 5)
- (2, 6)

- If the first die shows 3, the second die can show:
- (3, 1)
- (3, 2)
- (3, 3)
- (3, 4)
- (3, 5)
- (3, 6)

- If the first die shows 4, the second die can show:
- (4, 1)
- (4, 2)
- (4, 3)
- (4, 4)
- (4, 5)
- (4, 6)

- If the first die shows 5, the second die can show:
- (5, 1)
- (5, 2)
- (5, 3)
- (5, 4)
- (5, 5)
- (5, 6)

- If the first die shows 6, the second die can show:
- (6, 1)
- (6, 2)
- (6, 3)
- (6, 4)
- (6, 5)
- (6, 6)

### Step 4: Count the Outcomes
Each die has 6 outcomes, and since the two dice are independent, the total number of outcomes in the sample space is:
\[ 6 \times 6 = 36 \]

### Final Sample Space
Thus, the complete sample space can be represented as:
\[ S = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)\} \]