Answered

A pair of dice are rolled. Find each probability.

1. [tex]\( P(\text{sum is } 4) \)[/tex]

\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
1 & [tex]$(1,1)$[/tex] & [tex]$(1,2)$[/tex] & [tex]$(1,3)$[/tex] & [tex]$(1,4)$[/tex] & [tex]$(1,5)$[/tex] & [tex]$(1,6)$[/tex] \\
\hline
2 & [tex]$(2,1)$[/tex] & [tex]$(2,2)$[/tex] & [tex]$(2,3)$[/tex] & [tex]$(2,4)$[/tex] & [tex]$(2,5)$[/tex] & [tex]$(2,6)$[/tex] \\
\hline
3 & [tex]$(3,1)$[/tex] & [tex]$(3,2)$[/tex] & [tex]$(3,3)$[/tex] & [tex]$(3,4)$[/tex] & [tex]$(3,5)$[/tex] & [tex]$(3,6)$[/tex] \\
\hline
4 & [tex]$(4,1)$[/tex] & [tex]$(4,2)$[/tex] & [tex]$(4,3)$[/tex] & [tex]$(4,4)$[/tex] & [tex]$(4,5)$[/tex] & [tex]$(4,6)$[/tex] \\
\hline
5 & [tex]$(5,1)$[/tex] & [tex]$(5,2)$[/tex] & [tex]$(5,3)$[/tex] & [tex]$(5,4)$[/tex] & [tex]$(5,5)$[/tex] & [tex]$(5,6)$[/tex] \\
\hline
6 & [tex]$(6,1)$[/tex] & [tex]$(6,2)$[/tex] & [tex]$(6,3)$[/tex] & [tex]$(6,4)$[/tex] & [tex]$(6,5)$[/tex] & [tex]$(6,6)$[/tex] \\
\hline
\end{tabular}

[tex]\[
\frac{3}{36}
\][/tex]



Answer :

GinnyAnswer
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].