<s>[INST] Probability that the sum of three dice is not greater than 9 
What is the probability that the sum of 3 indistinguishable dice is less than or equal to 9?

I have tried counting the pairs $X_1, X_2$ such that their sum is less than or equal to 8, but I seem to be overcounting because I get that for each pair of dice I have 46 possibilities for their sum to be less than 8.
 [/INST] Consider the sum of the first two dice. If the sum is lower than or equal to 3, the third die can have any value. As the sum increases from 3 to 8, fewer possible values can be thrown by the third die. Since the number of ways to throw a sum of $n$ with two dice equals $6 - |n - 7|$, the number of ways to throw a total of 9 or less equals:
$$1 \cdot 6 + \sum_{i = 3}^{8} (6 - |i - 7|) (9 - i) = 1 \cdot 6 + 2 \cdot 6 + 3 \cdot 5 + 4 \cdot 4 + 5 \cdot 3 + 6 \cdot 2 + 5 \cdot 1 = 81$$
Since there are $6^3 = 216$ ways to throw the three dice, the probability of throwing 9 or less equals:
$$\frac{81}{216} = 0.375$$
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