Formulas

Combinatoris

Repetition Permutations Arrangement Combinations
Without $\\ P(m) = m!$ $\\ A_{m}^{n} = \frac{n!}{(n-m)!}; \; m\leqslant n$ $\\ C_{n}^{m} = \frac{n!}{(n-m)!\cdot m!}; \; m\leqslant n$
With $\\ P(m) = \frac{m!}{m_{1}!m_{2}!\: \ldots \: m_{k}!}$ WIP WIP

Law of total probability

$\\ P(A) = \sum_{i=1}^{n} P(H_{i})\times P(A|H_{i})$

Bernoulli formula

$\\ P_{n}(m) = C_{n}^{m}p^{m}q^{n-m}$

Poisson limit

$$\\ P_{n}(k)=\frac{\lambda ^{k}}{k!} e^{-\lambda} \\ \lambda = np$$

Integral Moivre–Laplace theorem

$$\\ P\left \{ k_{1} \leqslant k \leqslant k_{2} \right \} \approx \Phi(x_{2}) - \Phi(x_{1}) \\ \\ x_{1} = \frac{k_{1} - np}{\sqrt{npq}} \; \; ; \; \; x_{2} = \frac{k_{2} - np}{\sqrt{npq}} \\ \\ \Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{0}^{x}e^{-\frac{t^{2}}{2}} dt$$

see The students life task

Local Moivre–Laplace theorem

$$\\ P_{n}(k) \approx \frac{\varphi (x)}{\sqrt{npq}} \\ \\ x = \frac{k-np}{\sqrt{npq}} \; ; \; \varphi(x) = \frac{1}{\sqrt{2\pi }}e^{-\frac{x^{2}}{2}}$$

see Consignment with 15 defective products task
see R code