Intereting Posts

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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 :

- Binomial Distribution Problem - Airline Overbooking
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- The probability of a student speaking spanish is $30\%$. If we select $3$, what are the chances of at least one of them speaking Spanish?
- Finding mode in Binomial distribution

But if it is a binomial :

- What event do I regard as a success?
- What is its probability?
- 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.

- “8 Dice arranged as a Cube” Face-Sum Equals 14 Problem
- How to prove Poisson Distribution is the approximation of Binomial Distribution?
- Probability of rolling a die
- Binomial Distribution Problem - Airline Overbooking
- probability of rolling at least $n$ on $k$ 6-sided dice
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- What is the probability of the sum of four dice being 22?
- The probability of a student speaking spanish is $30\%$. If we select $3$, what are the chances of at least one of them speaking Spanish?
- What is a good strategy for this dice game?
- Sharper Lower Bounds for Binomial/Chernoff Tails

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 $P(X=0) > 0$.

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