Sure! Let's find the probability of rolling a sum of 4 with a pair of standard dice. Here is the step-by-step process:
1. Determine the total number of possible outcomes:
- When rolling two standard six-sided dice, each die has 6 faces. Therefore, the total number of possible outcomes is [tex]\( 6 \times 6 = 36 \)[/tex].
2. Identify the outcomes that result in a sum of 4:
- We need to list all the pairs [tex]\((\text{D}_1, \text{D}_2)\)[/tex] where the sum of the two dice is 4.
- These pairs are:
- [tex]\((1, 3)\)[/tex]: Die 1 shows a 1, and Die 2 shows a 3.
- [tex]\((2, 2)\)[/tex]: Die 1 shows a 2, and Die 2 shows a 2.
- [tex]\((3, 1)\)[/tex]: Die 1 shows a 3, and Die 2 shows a 1.
3. Count the number of favorable outcomes:
- From the list above, there are 3 favorable outcomes that produce a sum of 4.
4. Calculate the probability:
- The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes.
- Therefore, the probability [tex]\( P(\text{D}_1 + \text{D}_2 = 4) \)[/tex] is:
[tex]$
P(\text{D}_1 + \text{D}_2 = 4) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{3}{36} = \frac{1}{12}
$[/tex]
So, the probability of rolling a sum of 4 with two dice is:
[tex]$
P(\text{D}_1 + \text{D}_2 = 4) = \frac{1}{12} \approx 0.0833
$[/tex]
