<s>[INST] The odds in rolling two dice together. Two dice are rolled. I want to see the odds of the following:
$1.$ A sum of $5$.
$2.$ A sum of $8$ or $10$.
$3.$ A sum less than $6$.
$4.$ Not a sum of $7$.

Solution: $1.$ A sum of $5$. $5 = 1+4,2+3,3+2,4+1$. So the odds is $4/36 = 1/9.$
$2.$ A sum of $8$ or $10$. We can express $8 = 2+6,3+5,4+4,5+3,6+2$ and $10 = 4+6,5+5,6+4$. So the odds is $8/36 = 2/9$.
$3.$ A sum less than $6$. We can express $2=1+1$, $3=1+2,2+1$, $4=1+3,2+2,3+1$ and $5 = 1+4,2+3,3+2,4+1$. So the odds is $10/36 = 5/18$.
$4.$ Not a sum of $7$. We can express $7 = 1+6,2+5,3+4,4+3,5+2,6+1$. So the odds is $\frac{36-6}{36} = \frac{30}{36} = \frac56$

Is the solution correct?
 [/INST] As stated in the comments, you have correctly calculated the probabilities.
When two dice are rolled, there are $6 \cdot 6 = 36$ possible sums.  Of these, four outcomes yield a sum of $5$, while the remaining $36 - 4 = 32$ do not. Assuming each of the $36$ possible sums are equally likely to occur, the odds for a sum of $5$ being obtained is the ratio of the number of favorable events to the number of unfavorable events, which in this case is $4 : 32 = 1 : 8$.  The odds against a sum of $5$ being obtained is the ratio of the number of unfavorable events to the number of favorable events, which in this case, is $8 : 1$.

 Using the same reasoning, the odds for a sum of $8$ or $10$ is $8 : 28 = 2 : 7$ and the odds against the same outcome is $7 : 2$.  The odds for a sum less than $6$ is $10 : 26 = 5 : 13$ and the odds against the same outcome is $13 : 5$.  The odds for a sum other than $7$ is $30 : 6 = 5 : 1$ and the odds against the same outcome is $1 : 5$.

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