Currently reading through some basic probability theory, and have reached a chapter on binomial probability.
The name irks me, because my understanding is that it is essentially binary in nature (i.e. two possible outcomes).
Is it jsut that binomial the word used by mathematicians and binary the one used in software? Or am I missing something?
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$\begingroup$"Binomial" comes from the expansion of a two variable exponential expression, via the Binomial Theorem.
$$(x+y)^n = \sum_{k=0}^n \binom{n}{k} x^k y^{n-k}$$
The coefficient, $\binom nk$, is thusly known as the "binomial coefficient", and is the count of ways to select a subset of $k$ elements from a set of $n$ elements. It also counts distinct arrangements of a string of $k$ items of one type and $n-k$ items of another, when the items are considered indistinguishable except by type.
A Binomial random variable is the count of successes in a sequence of independent Bernoulli trials with identical success rate. When $n$ is the amount of trials and $p$ the success rate, then the probability mass function is:$$\mathsf P(X=k)=\dbinom nk p^k(1-p)^{1-k}\qquad\Big[k\in \{0,.., n\}\Big]$$
This is derived by counting distinct arrangements of a sequence of $k$ successes and $n-k$ failures and multiplying by the probability of obtaining any particular sequence.
As you can see, this is a term in the binomial expansion of $(p+(1-p))^n$, and indeed:
$$\sum_{k=0}^n \mathsf P(X=k) = \sum_{k=0}^n \dbinom nk p^k(1-p)^{1-k} = 1$$
PS: A Bernoulli trial is a single succeed-or-fail trial. The count of successes in a single Bernoulli trial is itself a Binomial Random Variable, and one which can realise only two values: $\{0,1\}$.
$\endgroup$ $\begingroup$To sum up Graham Kemp's nice and detailed explanation and give a to the point answer to your confusion:
Binary: Has two possible outcomes (e.g. 1/0, or flip of a coin)
Binomial: Count of outcomes in n binary trials (e.g. number of heads in 10 coin flips, number of 1's in a random binary number of length 8).
As you can see, a binomial random variable can have values other than 1/0, depending on its exact parameters. In the degenerate case where n=1 trial, a "binomial" is a "binary" (it is called a Bernoulli actually).
Disclaimer: The above is worded for intuitive understanding and not for mathematical rigor.
$\endgroup$ $\begingroup$The etymology of "binomial" is "having two parts", from bi- ("having two") and nomos (ancient Greek for "part, portion").
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