To determine the probability that the sum of the numbers on two fair 6-sided dice is greater than 10, we need to follow these steps:
1. Determine the total number of possible outcomes:
- Each die has 6 faces.
- When two dice are rolled, the total number of possible outcomes is [tex]\( 6 \times 6 = 36 \)[/tex].
2. Count the number of favorable outcomes:
- We need to find the combinations of dice rolls where the sum is greater than 10. Let's identify these combinations:
- If the sum of the dice is 11: possible combinations are (5, 6) and (6, 5).
- If the sum of the dice is 12: the possible combination is (6, 6).
- Therefore, there are 3 favorable outcomes in total:
[tex]\[
(5, 6), (6, 5), (6, 6)
\][/tex]
3. Calculate the probability:
- The probability [tex]\( P \)[/tex] of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
[tex]\[
P(\text{sum of two numbers on the dice is greater than 10}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} = \frac{3}{36}
\][/tex]
- Simplifying this fraction:
[tex]\[
\frac{3}{36} = \frac{1}{12}
\][/tex]
Therefore, the probability that the sum of the two numbers on the dice is greater than 10 is [tex]\( \frac{1}{12} \)[/tex].
