To find the probability of rolling a sum of eleven first and then a sum of twelve when a pair of dice is rolled twice, we need to approach the problem step-by-step.
### Step 1: Calculate the probability of rolling a sum of eleven
When rolling two dice, the pairs that will result in a sum of eleven are:
- (5, 6)
- (6, 5)
Each die has 6 faces, so the total number of possible outcomes when rolling two dice is [tex]\(6 \times 6 = 36\)[/tex].
Since there are 2 favorable outcomes out of 36 possible outcomes, the probability of rolling a sum of eleven is:
[tex]\[
P(\text{sum of 11}) = \frac{2}{36} = \frac{1}{18}
\][/tex]
### Step 2: Calculate the probability of rolling a sum of twelve
The pair that will result in a sum of twelve when rolling two dice is:
- (6, 6)
Since there is only 1 favorable outcome out of 36 possible outcomes, the probability of rolling a sum of twelve is:
[tex]\[
P(\text{sum of 12}) = \frac{1}{36}
\][/tex]
### Step 3: Calculate the combined probability
To find the combined probability of rolling a sum of eleven first and then a sum of twelve, we multiply the probabilities of the individual events (since they are independent events).
Thus, the combined probability is:
[tex]\[
P(\text{sum of 11 first and sum of 12 second}) = P(\text{sum of 11}) \times P(\text{sum of 12}) = \frac{1}{18} \times \frac{1}{36} = \frac{1}{648}
\][/tex]
Therefore, the probability of rolling a sum of eleven first and then a sum of twelve is:
[tex]\[
\boxed{\frac{1}{648}}
\][/tex]
