# Why is the sum of the rolls of two dices a Binomial Distribution? What is defined as a success in this experiment?

I know that a Binomial Distribution, with parameters n and p, is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p.

I read that when the sum of the roll of two dices is a binomial distribution. Is this right? I know that the sum looks like a binomial :

But if it is a binomial :

1. What event do I regard as a success?
2. What is its probability?
3. How many times do I repeat the experiment?

I mean a Binomial Distribution measures the probability of observing an event ( which has probability of success p), k times in specific amount of repetitions n. Here I see 12 different events with different probabilities and I am confused.

#### Solutions Collecting From Web of "Why is the sum of the rolls of two dices a Binomial Distribution? What is defined as a success in this experiment?"

TLDR: The sum of two $n$-sided dice is not binomially distributed.

A discrete random variable, $X$, has a binomial distribution, $X\sim Bin(n,p)$ when $Pr(X=x) = \begin{cases}\binom{n}{x}p^x(1-p)^{n-x}&\text{for}~x\in\{0,1,2,\dots,n\}\\ 0 & \text{otherwise}\end{cases}$

For $X$ the sum of two $n$-sided dice however, $Pr(X=x) = \begin{cases} \frac{n – |x-(n+1)|}{n^2} & \text{for}~x\in\{2,3,\dots,2n\}\\ 0 & \text{otherwise}\end{cases}$

Notice that since $n$ will be a fixed number, $Pr(X=x)$ is linear on the intervals $[2,n+1]$ and again linear on the intervals $[n+1,2n]$. This is in direct contrast to the binomial distribution scenario where $Pr(X=x)$ is definitely not linear (as it has terms like $p^x$ and $\binom{n}{x}$ appearing in the formula).

As mentioned in the comments above, as $n$ grows large, the histogram for the sum of two $n$-sided dice approaches the shape of a triangle.

This becomes even more apparent as $n$ gets even larger. Here is the start of the histogram for $n\approx 30$ (its a lot of effort to complete, but you get the idea).

On the other hand, the binomial distribution appears with the all-familiar “bell-shaped” curve.

As such, these are two very different distributions and should not be confused.

No, this is not a binomial distribution. For one thing, a binomial distribution would have $P(X=0) > 0$.