To find the probability of rolling a sum of 4 with two standard dice, let's follow a step-by-step approach:
1. Determine the total number of possible outcomes:
Each die has 6 faces (1 through 6). Hence, when rolling two dice, the total number of possible outcomes is calculated by multiplying the number of faces of the first die by the number of faces of the second die.
[tex]\[
\text{Total outcomes} = 6 \times 6 = 36
\][/tex]
2. List the favorable outcomes for the sum of 4:
To find the sum of 4, we need to determine the pairs of numbers on the two dice that add up to 4. These pairs are:
- (1, 3)
- (2, 2)
- (3, 1)
Each of these pairs represents a favorable outcome.
3. Count the number of favorable outcomes:
There are three pairs that add up to 4, so the number of favorable outcomes is:
[tex]\[
\text{Favorable outcomes} = 3
\][/tex]
4. Calculate the probability:
The probability [tex]\(P\)[/tex] of rolling a sum of 4 is the ratio of the number of favorable outcomes to the total number of possible outcomes.
[tex]\[
P(D_1 + D_2 = 4) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{3}{36}
\][/tex]
5. Simplify the fraction:
The fraction [tex]\(\frac{3}{36}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
[tex]\[
P(D_1 + D_2 = 4) = \frac{3 \div 3}{36 \div 3} = \frac{1}{12}
\][/tex]
Therefore, the probability of rolling a sum of 4 with two standard dice is:
[tex]\[
P\left(D_1 + D_2 = 4\right) = \frac{1}{12}
\][/tex]
