To find the probability that the total showing on two rolled dice is either even or less than nine, we start by analyzing the scenario carefully.
### Step-by-Step Solution:
1. Total Number of 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. Outcomes When the Total is Even:
- Two dice can sum up to an even number if the sum of the two numbers is 2, 4, 6, 8, 10, or 12.
- The even totals can be formed as follows:
- Sum of 2: (1,1)
- Sum of 4: (1,3), (2,2), (3,1)
- Sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1)
- Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2)
- Sum of 10: (4,6), (5,5), (6,4)
- Sum of 12: (6,6)
- Counting these outcomes, there are 18 outcomes where the sum is even.
3. Outcomes When the Total is Less Than Nine:
- Totals of 2, 3, 4, 5, 6, 7, or 8 are all totals less than nine.
- These totals can be formed as follows:
- Sum of 2: (1,1)
- Sum of 3: (1,2), (2,1)
- Sum of 4: (1,3), (2,2), (3,1)
- Sum of 5: (1,4), (2,3), (3,2), (4,1)
- Sum of 6: (1,5), (2,4), (3,3), (4,2), (5,1)
- Sum of 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
- Sum of 8: (2,6), (3,5), (4,4), (5,3), (6,2)
- Counting these outcomes, there are 26 outcomes where the sum is less than nine.
4. Combining Both Conditions:
- To find the number of outcomes where the sum is either even or less than nine, we need to combine these two sets of outcomes without double-counting any outcomes.
- We combine the favorable outcomes:
- Outcomes that are either even or less than nine or both.
- Counting these outcomes, there are 30 favorable outcomes where either condition is met.
5. Calculating the Probability:
- The probability is the number of favorable outcomes divided by the total number of possible outcomes.
- So, the probability is [tex]\(\frac{30}{36} = \frac{5}{6}\)[/tex].
Thus, the probability that the total showing is either even or less than nine is:
[tex]\[ \boxed{\frac{5}{6}} \][/tex]
