To find the probability of rolling a sum of 11 with a pair of standard six-sided dice, follow these steps:
1. Identify the possible outcomes for each die:
Each die can land on any one of the six faces. We’ll list these faces as {1, 2, 3, 4, 5, 6}.
2. Consider all possible pairs of outcomes from the two dice:
Since each die has 6 faces, there are [tex]\(6 \times 6 = 36\)[/tex] possible outcomes when rolling two dice.
3. Determine the number of successful outcomes:
We need to find pairs [tex]\((d1, d2)\)[/tex] such that the sum [tex]\(d1 + d2 = 11\)[/tex]. Checking all combinations:
- If [tex]\(d1 = 5\)[/tex], then [tex]\(d2\)[/tex] must be 6 (as [tex]\(5+6=11\)[/tex]).
- If [tex]\(d1 = 6\)[/tex], then [tex]\(d2\)[/tex] must be 5 (as [tex]\(6+5=11\)[/tex]).
So the successful outcomes where the sum is 11 are:
[tex]\[
(5, 6) \text{ and } (6, 5)
\][/tex]
There are exactly 2 successful outcomes.
4. Calculate the probability:
The probability [tex]\(P(\text{sum of 11})\)[/tex] is given by the ratio of the number of successful outcomes to the total number of outcomes:
[tex]\[
P(\text{sum of 11}) = \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Outcomes}} = \frac{2}{36}
\][/tex]
5. Reduce the fraction:
Simplify [tex]\(\frac{2}{36}\)[/tex] to its lowest terms:
[tex]\[
\frac{2}{36} = \frac{1}{18}
\][/tex]
Therefore, the probability of rolling a sum of 11 with a pair of standard dice is:
[tex]\[
\boxed{\frac{1}{18}}
\][/tex]
