Answer :
To solve for the probability that the sum of two rolled dice is 4, follow these steps:
### Step-by-Step Solution
1. Determine the total number of possible outcomes:
- Each die has 6 faces.
- When rolling two dice, each die's outcome is independent of the other.
- Therefore, the total number of possible outcomes is [tex]\(6 \times 6 = 36\)[/tex].
2. Identify successful outcomes that result in a sum of 4:
- We need to find all pairs of dice rolls [tex]\((x, y)\)[/tex] such that [tex]\(x + y = 4\)[/tex].
- Let's enumerate these pairs:
- [tex]\((1, 3)\)[/tex]: 1 on the first die and 3 on the second die.
- [tex]\((2, 2)\)[/tex]: 2 on the first die and 2 on the second die.
- [tex]\((3, 1)\)[/tex]: 3 on the first die and 1 on the second die.
3. Count the number of successful outcomes:
- From the enumeration above, there are 3 successful outcomes: [tex]\((1, 3), (2, 2), (3, 1)\)[/tex].
4. Calculate the probability of rolling a sum of 4:
- Probability [tex]\(P(\text{sum is 4})\)[/tex] is given by the ratio of the number of successful outcomes to the total number of possible outcomes.
- Thus, [tex]\(P(\text{sum is 4}) = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{3}{36}\)[/tex].
5. Simplify the fraction (if necessary):
- [tex]\(\frac{3}{36}\)[/tex] simplifies to [tex]\(\frac{1}{12}\)[/tex].
6. Expressing the answer as a decimal:
- [tex]\(\frac{1}{12} = 0.08333333333333333\)[/tex].
### Final Answer
The probability that the sum of two rolled dice is 4 is [tex]\( \frac{1}{12} \)[/tex] or approximately [tex]\(0.0833\)[/tex].
### Step-by-Step Solution
1. Determine the total number of possible outcomes:
- Each die has 6 faces.
- When rolling two dice, each die's outcome is independent of the other.
- Therefore, the total number of possible outcomes is [tex]\(6 \times 6 = 36\)[/tex].
2. Identify successful outcomes that result in a sum of 4:
- We need to find all pairs of dice rolls [tex]\((x, y)\)[/tex] such that [tex]\(x + y = 4\)[/tex].
- Let's enumerate these pairs:
- [tex]\((1, 3)\)[/tex]: 1 on the first die and 3 on the second die.
- [tex]\((2, 2)\)[/tex]: 2 on the first die and 2 on the second die.
- [tex]\((3, 1)\)[/tex]: 3 on the first die and 1 on the second die.
3. Count the number of successful outcomes:
- From the enumeration above, there are 3 successful outcomes: [tex]\((1, 3), (2, 2), (3, 1)\)[/tex].
4. Calculate the probability of rolling a sum of 4:
- Probability [tex]\(P(\text{sum is 4})\)[/tex] is given by the ratio of the number of successful outcomes to the total number of possible outcomes.
- Thus, [tex]\(P(\text{sum is 4}) = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{3}{36}\)[/tex].
5. Simplify the fraction (if necessary):
- [tex]\(\frac{3}{36}\)[/tex] simplifies to [tex]\(\frac{1}{12}\)[/tex].
6. Expressing the answer as a decimal:
- [tex]\(\frac{1}{12} = 0.08333333333333333\)[/tex].
### Final Answer
The probability that the sum of two rolled dice is 4 is [tex]\( \frac{1}{12} \)[/tex] or approximately [tex]\(0.0833\)[/tex].
