Is it appropriate use the Binomial Theorem to analyze the problem of rolling dice?

Question

In mathematics, the multinomial theorem

describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials.

which means not each multinomial distribution is necessarily a binomial distribution, such as rolling dice.

this post is using the Binomial Theorem to analyze the problem of rolling dice, is it appropriate?

in other words, is rolling dice a multinomial distribution that is not a binomial distribution?

Answer 1

This really depends on exactly what you are looking at. "Rolling dice" is a specification of an activity, not a specification of a numerical outcome that constitutes a random variable having a distribution. If you roll a set of standard six-sided dice and you look at the counts of the six possible outcomes, then (under standard assumptions), this vector of count outcomes will have a multinomial distribution. On the other hand, if you look at the count of only one outcome, then (under standard assumptions), this value will have a binomial distribution. There are many other distributions you could get from "rolling dice", depending on what numerical outcome you look at.

Answer 2

I think your question's confusion stems from interpreting "rolling dice" as a "problem." Rolling dice is just rolling dice.

The binomial distribution is useful for describing the probabilities of a number of independent events occurring $x$ times out of $n$ attempts, each with probability $p$. If "an event" is rolling some number, say "rolling a 3" or "rolling a 4 or higher", then the binomial theorem can help you understand the probabilities of such events occurring. For example: what is the probability that a fair¹ six-sided² die will come up '3' exactly four times if I roll the die ten times?

$$\begin{array}{rcl} x & = & 4\\ n & = & 10\\ p & = & \frac{1}{6}\\ P\left(X = 6|n,p\right) & = & {10! \choose 4!\left(10-4\right)!}\left(\frac{1}{6}\right)^{4}\left(1-\frac{1}{6}\right)^{10-4}=0.054\end{array}$$

By contrast the multinomial theorem gives a means of calculating the number of unique ways of ordering $n$ total things across $i$ types of those things, where the number of the first type of thing is $n_{1}$, the number of the second type of thing is $n_{2}$, etc. up to $n_{i}$. For example, suppose I roll 10 six sided dice, and obtain $5,6,5,1,5,6,3,2,1,2$, I have three '5's, two 'six's, two '1's, one '3' and two '2's, and I want to know how many unique ways I could have rolled that many '5's, '6's etc.:

$$\text{Unique permutations of $n$ rolls with above $i$ values} = {10! \choose 3!\times 2! \times 2! \times 1! \times 2!} = 75,600$$

So you can see that "rolling dice" is not a "problem", or at least not until we ask specific questions about particular kinds of behavior of dice rolls, and then the binomial distribution, or multinomial theorem, or some other statistical or mathematical tool may be useful for providing answers.

¹ "Fair" means each side has an equal probability of coming up on a single roll.

² There are dice with different numbers of sides.