how to visualize sample space in exercises like "roll 3 6-sided dice, and find probability that sum is $\geq$, $\leq$, or $x$? if I have three 6-sided dice, and I need to find probability that sum is $x$, $\leq$, or $\geq x$, under these conditions I need to sketch sample space in order to find events I'm interested in. This would be simple if I'd have to roll two 6-sided dice, because I simply have to write each number of the first die vertically and each number of the second die horizontally (to visualize it, think of microsoft excel in which you draw a table with rows, and columns), and this is the best way to visualize sample space. But here's the problem: what if I have 3 dice? it would be tedious to write each event without creating a table, because I'd have more than two dice. Where should I put the third die? I was wondering if there's a way to visualize these kind of problems easily.
 A: One method: You could do something like this: Start with a table of two dice sums:
$$\begin{array}{c|cccccc} + & 1&2&3&4&5&6 \\ \hline 
1&2&3&4&5&6&7 \\
2&3&4&5&6&7&8 \\
3&4&5&6&7&8&9 \\
4&5&6&7&8&9&10 \\
5&6&7&8&9&10&11 \\
6&7&8&9&10&11&12
\end{array}$$
Then for any $k$ with $3\le k \le 18$, you could look at this table and determine how many ways you could get $k$ from a table entry with one more die roll.
For instance if $k=9$, any entry in the table from $3$ to $8$ could get to a sum of $9$ with the roll of a third die. There are $25$ entries in this table in that range. So there are $25$ ways to roll three dice to a sum of $9$.
Another method: Make six tables, each six by six (so each table represents rolling two of the three dice). Each table is associated with a die roll ($1, 2, 3, 4, 5,$ or  $6$). Then at each entry of a given table, add the two table indices plus the associated third roll.
For instance the table where the third roll is $4$ would look like this.
$${\rm Third\; roll\; is\;} 4 \hspace{.2in}  \begin{array}{c|cccccc} 4+ & 1&2&3&4&5&6 \\ \hline 
1&6&7&8&9&10&11 \\
2&7&8&9&10&11&12 \\
3&8&9&10&11&12&13\\
4&9&10&11&12&13&14 \\
5&10&11&12&13&14&15 \\
6&11&12&13&14&15&16
\end{array}$$
The six tables constructed in this way would give you a diagram of the full sample space.
