Counting & Binomial Distribution

Counting

$nn$ is any non-negative integer and $0\le k\le n0\; \backslash leq\; k\; \backslash leq\; n$

✔️ Combinations

• A collection of the elements where the order doesn’t matter

>>> AB AC AD BC BD CD

✔️ Combinations with Replacement

• This method takes under consideration the combination of a number with itself as well.

>>> AA AB AC AD BB BC BD CC CD DD

✔️ Permutations

• An arrangement of a set where the order does matter.

>>> AB AC AD BA BC BD CA CB CD DA DB DC

✔️ Product

• Cartesian product of input iterables

>>> AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD

Binomial Distribution

Let $XX$ be the random variable counting the number of hits in a Bernoulli scheme at $nn$ repetitions.

The law of $XX$ is the binomial law of parameters $nn$ and $pp$ and we have:

• $p(X = k) = \left( {n\atop k} \right)p^k(1 - p)^{n-k}$

Where $nn$ is any non-negative integer and $0\le k\le n0\; \backslash leq\; k\; \backslash leq\; n$

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✔️ Mean

• $n\times pn\; \backslash times\; p$

✔️ Variance

• $n\times p\times (1-p)n\; \backslash times\; p\; \backslash times\; (1\; -\; p)$

✔️ Standard Deviation

• $\sqrt{n\times p\times (1-p)}\backslash sqrt\{n\; \backslash times\; p\; \backslash times\; (\; 1\; -\; p)\}$