How do I represent the probability of a particular die roll in formal notation?

Question

I'm trying to represent the projected outcome of a series of die rolls in a formal, nice-looking notation.

The die rolls go as follows:

1. Roll a two-sided die twice and then add two to the sum of the output. This represents the number of objects generated.
2. For each generated object, roll a two-sided die to determine the value of each object.
3. The total output is the sum of values.

I've made a table of what the probability of different outcomes would be, but I want to include a title over the graph and I'm not familiar with probability notation and don't know how to represent what I have.

P(?) = (2d2 + 2)d2

How do I represent the right-hand-side of the equation in formal probability notation?

Answer 1

What you've defined is a "random variable".

The usual notation for one is an upper-case italic, like $X$, $Y$, or $X_i$ if you've got multiple ones. If there is a meaningful name for the quantity, like say "stealth", then $X_{stealth}$ might be used.

A discrete random variable like this is usually presented in a table like this (which it sounds like you've already got):

\begin{array} {|r|r:r:r:r:r|l|} \hline X_{foo} & -2 & -1 & 0 & 1 & 2 \\ \hdashline \mathsf P(X_{foo}=x) & 0.1 & 0.2 & 0.2 & 0.2 & 0.3 \\ \hline \end{array}

(I assume your random variable can assume many more values than this example, so you would probably use two columns instead of two rows. That's fine, this was just the first snippet of LaTeX code I found.)

As for the title, the (standard?) dice notation runs a bit contrary to standard random variable notation. When you write "2d6" you mean rolling two separate dice, but $2X_{d6}$ would mean roll one and double the value generated - "$X_1 + X_2$, where $X_i \sim d6$" would be appropriate random variable notation.

I'd say, since you're already assuming familiarity with dice notation, that you stick with it, and a title mostly like what you already have would work fine:

$$X_{foo} = (2d2 + 2)\cdot d2$$

The fact that you're using dice rolls to determine how many dice to roll in the second stage feels a little non-standard to me, so hopefully you'd have already explained your procedure before presenting the table. But with that done I think your notation is clear.

(If you want the fully math-y way to write this out, I could provide it, but I think it'd actually just be confusing in your case. If you're already using the "2d6" notation, why get pedantic at this stage?)