To find the probability of rolling a sum of 3 with a pair of standard dice, we can follow these steps:
### Step-by-Step Solution:
1. Determine the Total Number of Possible Outcomes:
When two dice are rolled, each die has 6 faces. Therefore, the total number of possible outcomes is:
[tex]\[
6 \times 6 = 36
\][/tex]
2. Identify the Favorable Outcomes:
We need to find all possible pairs of dice rolls that sum up to 3. Let's list them:
- If the first die shows a 1, the second die must show a 2. This gives us the pair (1, 2).
- If the first die shows a 2, the second die must show a 1. This gives us the pair (2, 1).
Thus, there are 2 favorable outcomes: (1, 2) and (2, 1).
3. Calculate the Probability:
The probability [tex]\( P \)[/tex] of rolling a sum of 3 is the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[
P\left(D_1 + D_2 = 3\right) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}}
\][/tex]
Therefore,
[tex]\[
P\left(D_1 + D_2 = 3\right) = \frac{2}{36} = \frac{1}{18}
\][/tex]
So, the probability of rolling a sum of 3 with a pair of standard dice is:
[tex]\[
P\left(D_1 + D_2 = 3\right) = \frac{1}{18}
\][/tex]
