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$$P_n = n! = 1\cdot 2 \cdot 3 \cdot ... \cdot (n-1) \cdot n$$

$$A_m^n = n \cdot (n-1) \cdot ... \cdot (n-m+1)$$

$$C_n^m =\frac{A_n^m}{P_m}=\frac{n!}{m! \cdot (n-m)!}$$

$$ P_n (n_1,n_2,...,n_k)=\frac{n!}{n_1! \cdot n_2!\cdot ... \cdot n_k!}. $$

$$\overline{A}_n^k=n\cdot n\cdot ... \cdot n = n^k. $$

$$\overline{C}_n^k=C_{k+n-1}^k=\frac{(k+n-1)!}{(n-1)!\cdot k!}$$

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$$P(A) = \frac{m}{n},$$ $m$ - $A$ , $n$ - .

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$$ P(A+B) = P(A)+P(B) $$

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$$ P(A+B) = P(A)+P(B)-P(AB) $$

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$$ P(A\cdot B) =P(A)\cdot P(B) $$

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$$ P(A\cdot B) =P(A)\cdot P(B|A),\\ P(A\cdot B) =P(B)\cdot P(A|B). $$

$P(A|B)$ - $A$ , $B$,

$P(B|A)$ - $B$ , $A$.

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$$ P(A)=\sum_{k=1}^{n} P(H_k)\cdot P(A|H_k), $$

$H_1, H_2, ..., H_n$ - .


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$$ P(H_m|A) =\frac{P(H_m)\cdot P(A|H_m)}{P(A)} = \frac{P(H_m)\cdot P(A|H_m)}{\sum\limits_{k=1}^{n} P(H_k)\cdot P(A|H_k)}, $$

$H_1, H_2, ..., H_n$ - .

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$$ P_n(k)=C_n^k \cdot p^k \cdot (1-p)^{n-k} = \frac{n!}{k! \cdot (n-k)!}\cdot p^k \cdot (1-p)^{n-k} $$ $k$ $n$ , $p$ - .

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$k_0$ $n$ ( $p$ - ):

$$ np-(1-p) \le k_0 \le np+p. $$

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$n$ , $p$ , $np \lt 10$, :

$$ P_n(k)=\frac{\lambda^k}{k!}\cdot e^{-\lambda}. $$

$\lambda=n \cdot p$ .

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$$ P_n(k) = \frac{1}{\sqrt{npq}} \varphi\left( \frac{k-np}{\sqrt{npq}} \right) $$

$k$ $n$ , $p$ - , $q=1-p$.
$\varphi(x)$ .


$$ P_n(m_1, m_2) = \Phi\left( \frac{m_2-np}{\sqrt{npq}} \right)-\Phi\left( \frac{m_1-np}{\sqrt{npq}} \right) $$

$m_1$ $m_2$ $n$ , $p$ - , $q=1-p$.
$\Phi(x)$ .

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$$ P\left( \left| \frac{m}{n} -p\right| \le \varepsilon\right) = 2 \Phi\left( \varepsilon\cdot \frac{\sqrt{n}}{\sqrt{p(1-p)}} \right) $$

$\varepsilon$ - , $p$ - .

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