how to visualize sample space in exercises like "roll 3 6-sided dice, and find probability that sum is $\geq$, $\leq$, or $x$?

Question

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.

Answer 1

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.