??? ?? ???????
????? ????????
?????? ????????
???? ??????? ???? ?????? ??? ???? ??? ????? ?????? ???? ????????
????? ?????? ?????? ????? ?????? ????? ?????? ????
??????? ??????? ???? ????? ???????? ????? ????? ??????? ??????? ?????? ???? ??????? ???
???? ?? ???? ??????????
????? ???????
? ? ?

????? ???????? ( ??????????? : Probability theory )? ?? ??????? ???? ???? ?????? ??????? ?????????? ??????? ?????????? ?????????? ????? ?????? ?? ?????? ??? 0 ?1 ???? ?????? ???? ?? ??? ???? ??? ???? ?????? ?? ??? ????. ^{[1] [2] [3]}

??? ????? ?????? ????? ${\displaystyle E}$ ??????? ${\displaystyle P(E)\!}$ ??? ?????? ???????? . ??? ???? ?????? ????? ${\displaystyle E}$ ???? ????? ????? ${\displaystyle F}$: ???????? ?????? ????? ${\displaystyle E}$ ?? ????? ????? ${\displaystyle F}$. ???? ??? ???????? ?????? ??????? ??? ?????? ??????? ??? ??????? (?? ??????? ???) ??? ?????? ???? ????? ${\displaystyle F}$? ?? ${\displaystyle P(E\cap F)/P(F)}$.

??? ?? ????? ???? ???????? ?????? ????? ${\displaystyle E}$ ???? ????? ${\displaystyle F}$ ?? ?????? ??????? ??? ??????? ????? ?? ?? ???????? ???? ?? ??? ???? ${\displaystyle F}$ ?? ??? ????? ????? ???? ?? ???? ??????? ???????.

????? ????? ?????????? ??????? ???? ?? ??????? ????: ??????? ???????? ???????? ????????? ??????? ????????.

??????? [ ??? ]

???? ?????????? ???????? ?? ??????? ?????? ?? ??? ??????? ????? ????? ??????? ???? ??????? ? ??? ??????? ?????? ??????? ??? ??????????. ?????? ?? ???? ????????? ??? ????? ?? ??? ???????.

????? ??????? ??????????? ?????????? ?????? ?? ??????? ??? ?????? ???? ???? ?? ??? ???????? ??????? ???? ??? ???? ????? ?????? ??? ???????? ??? ??? ???? ?? ????? ????? ?????? ? ?????? ???? ???? ????? ?????? ??? (???? ??? ???? ?????? ??? ????? ????? ). ???? ??? ???????? ????? . ?????? ?????????? ???? ?????? ?????? ?????? ???? ?????? ?? ????? ?????? ??? ? ? ?? ??????? ????? ???? ?????????? ?? ???? ????? ????? ????? ?? ??? ?????? ????? ?? ??????? ??????. ?????????? ?? ??? ??????? ????? ???? ?????????? ???????? ????????? ???????? ????????. ??? ???? ??? ??? ?? ??? ??????? ????? ???? ?????? ?????. ??????? ?????????? ????? ????? ?????? ???? ???? ??????? ?? ???????? ?????? ????????.

???? ???? ??????? [ ??? ]

???? ????? ?????????? ?????? ??????? ?????????? ?? ??????? ???????? ?????? ???????? ?? ??????? ?????????? ????????? ????????? ????????? . ??? ??? ?????????? ????? ????? ????? ????? ????? ?????????? ????? ??? ???? ?????????? ???? ????? ??????? ??? ????? ???? ????? ?????? ???? ?? ??? 1933 ???? ??????? ?????? ?????????? ?? ??? ?? ???? ??? ????? ?????? ??????? ?????? ?????????? ???????.

???? ????? ?????? ????????? ??? ??? ?????? ${\displaystyle (\Omega ,{\mathcal {F}},P)}$, ???

• ${\displaystyle \Omega }$ ???? ?????? ??? ?????? ???? ???? ?????? .
• ${\displaystyle {\mathcal {F}}}$ ?? σ-??? ????? ?????? ???? ???? ?? ???? ?? ??????? : ≪???≫.

??? ?????? ?? ???? ?? ${\displaystyle {\mathcal {F}}}$ ???? ?????-??? ??? ????? ???????? ???? ???? ${\displaystyle \Omega }$, ??? ????? ?? ??? ???? ???? ????? ??????? ?? ????? ????? ?? ??? ????.

• ${\displaystyle P}$ ???? ???? ??????? probability measure ??? ${\displaystyle {\mathcal {F}}}$, ??, ???? ???? ????

${\displaystyle P(\Omega )=1}$, ?? ?? ?????? ???? ???? ?????? ????? ??????. ???? ???????? ${\displaystyle (\Omega ,{\mathcal {F}})}$ ???? ????? ?? ???? ????? ?????? ???? ????? ??? ???? ??????? ?????? ???? ??????? ????. ?? ????? ?? ????? ?? ${\displaystyle P}$ ???? ???? ????? ??? ${\displaystyle {\mathcal {F}}}$ ???? ??? ???? ?????? ${\displaystyle \Omega }$.

????? ???????? [ ??? ]

?? ??? ?????????? ??????????? ? ????? ???????? ( ??????????? : Probability distribution )? ?? ????? ?????? ???? ??? ?????? ????? ????? ?????? ?? ?????? ????? ????? ??????? ??. ??????? ???? ?? ???? ??????? ????? ????? ??? ????? ??? ?????? ??????? ???????? . ^[4]

??????? ????????? ????? ???? ???? ?? ????? ???? ?????? ?? ?????? ?????????? ???? ????? ???? ???? ??? ???????? ???????? ????? ?? ?????? ?????? ???? ???? ?????? ??????????.

?? ????? ?????? ??????? ????? ??????? ????? ???? ????????? ?????? ?? ??? ???????. ???? ??? ??????? X ?????? ??????? ??? ??????? ????????? ??????? ?? ???? ?????? [ a , b ] ??????? : ????? ?? ?????? ?? ???? ??????? ${\displaystyle X}$ ???? ??? ?????? ?? : ${\displaystyle \Pr \left[a\leq X\leq b\right]}$.

????? ????????? ????????? [ ??? ]

?? ????? ??????????? ???? ??? ?????? ?????? ? ????? ????????? ????????? . ??? ????? ????? ????????? ????????? ???? ????????? ????????? ???? ??????? ????? ?? ????? ?????????? ????????? ??? ??????? ????????? ????????? . ????? ??? ???????? ?? ????????? ?????? ?????? ???? ??????? ???????? ????? ?????? ?????? ??? ???? ??? ???? ???? ????? ??????? ????????? ?? ??? ???????? ?? ??? ??????? ?? ???? ?? ????? ???? ????? ??? ????? ??????? ??????? ????? ????? ?? ???????. ????? ???????? ???????? ??????? ??????? ?????? ??? ??? ??? ??????: ?????? ?????? ?????? ??? ?? ??????? ???? ?? ????? ?????? ???? ?????? ??? ????? ?? ??????? ????? ?? ??????? ???? ???? ????? ?????? ?????????? ??? ???????. ^[5]

????? ??????? ??????? [ ??? ]

???? ????? ??????? ??????? ??? ?????? ?????? ?????? ????? ????? ???? ????? ?? ???????? ?????? ?? ?????? ??? ???? ????? ????? ????? ?

?????? ??????? ???????? [ ??? ]

?? ????? ????????? ???? ??????? ??????? ???????? ( ??????????? : Central limit theorem )? ?????? ????? ?????? ?????????? ??? ?? ????? ??? ??????? ??????? ???????? ?????? ? ???? ??? ?????? ??? ????? ??????? ????. ^{[6] [7] [8]}

??? ??? ????????? ???? ??? ??? ???? ????????? ???????? ???? ??????? ????? ??? ??????? ???? ??? ?????? ?????? ?? ??? ???? ?????? ???????? ??????.

???? ?????? ??????? ???????? ???? ????????? ???????? ??????? ?? ???????.

???? X1, X2, X3,... Xn ??????? ?? ??????????????? ?????????? ?? ???????

??????? ???????? ??? ???? ???? ???? ????? ????? μ ???????? σ2 > 0.

???? ?????? ??????? ???????? ?? : ???? ????? ??? ?????? n ,??? ??????? ?????? ??? ????????? ????????? ????? ?? ??????? ??????? ???????.

????? ???????? [ ??? ]

?? ??????? ???? ??? ?? ???? ??? ??? ???? ??????? ????? ???????? ????? ??????? ?? ?????? ??????? ??????? ???????? ???? ??? ?? ??? ??????? ???? ???????? ???????? ??????? ?? ????? ??? ????? ?????? ????????? ??? ??????? ??????? ??????? ??? ??????? ?????? ???? ???????? ??? ????? ??????? ?????? ?????? ???????? ???????? ?? ??? ?????? ?????? ???????? ?????? ????????? ???? ?? ????? ????? ???? ?????? ??? ??????? ?????? ????????? ???? ??????? ?? ????? ???????? ??????? ??:

1. ???? ???????? ??????? ???????? ??????.
2. ???? ???????? ?????? ???????? ???? ?????? ?? ???? ?????? ??? ??????? ???????? ?????? .
3. ??? ????? ??????? ?????????? ??????????.

????? ???????? [ ??? ]

???????? ??????? [ ??? ]

??? ????? ???????? ????? ??????? ??????? ?????? ??? ?? ??? ??? ????? ??? ????? ?? ${\displaystyle 1/6}$

???????? ?????? ?? ?????? (Subjective Probabilities) [ ??? ]

???????? ???? ?????? ??? ??? ??? ???? ????? ?? ??????? ??? ??????? ??? ????? ?? ??? ???? ??????? ??? ???? ?? ???? ?????.

?????????? ????????? ??????? (The Relative Frequency) [ ??? ]

???? ?????? ??? ???:

?) ???? ???? ????? ??? ??? ???? ?? ???? ?????? ??????? ??????.

?) ???? ???? ????? ?? ??? ???? ?? ????????? ?? : ??? ???? ????? ????? ??? ??? ???? ????? ??????? .

???????? ???????? ???????? [ ??? ]

??????? ????????? (RANDOM SAMPLING): ?? ????? ???? ?? ???? ??????? ??? ????? ?? ???? ????? ????? ?? ??? ????? ???????? ????????? ??? ?? ????? ???? ?????? ?????? ??. ??????? ????????? ?????? ???? ?????? ???? ??????? ???????? {????? ?????} ??? ??? ?? ???? ????? ?????? ???????? ?????? ??????. ??????? ????????? ?????? ??? ????? ???? ?????? ???????? ${\displaystyle \{1,2,3,4,5,6\}}$ ??? ??? ?? ????? ????? ...

???? ??????? (Sample Space) [ ??? ]

???? ???????? ${\displaystyle \{1,2,3,4,5,6\}}$ ?? ?????? ?????? ??????? ????????? ????? ??????? ?? ???? ?????????? ?? ???? ?????? (Sample Space) ???? ?????? ?????? ????? ???? ???? ??? ????? ${\displaystyle \{T,H\}}$ ?? ???? ???? ??? ?????? ?? ????? ??????? ??????? ?????? ???????? ?????????.

??????? [ ??? ]

????? ?? ?????? ????? ?? ???? ?????? ???? ??????? ???? ?????? ${\displaystyle N^{2}}$ ??? ${\displaystyle N}$ ??? ????? ???? ?????? ??????? ???? ????? ${\displaystyle A}$ ?? ???? ??? ????? ????? ?????? ??????? ??? ??????? ??????? ?????? ?? ??: ${\displaystyle P(A)=M/N}$ ??? ${\displaystyle M}$ ??? ????? ???? ${\displaystyle A}$ ??????? ${\displaystyle N}$ ??? ??????? ??????? ??????? ???? ??? ???? ??? ????? ??? ????? ??? ????? ?? ${\displaystyle 0.5}$ ??? ??????? ??????? ????? ${\displaystyle (1,3,5)}$ ????? ???? ??????? (??? ????) ??? ??????? ??? ${\displaystyle (1,2,3,4,5,6)}$ ????????? ${\displaystyle 3/6=0.5}$ ? ????? ??????? ???? ????? ?? ????? ?? ??????.

????? ?????? ( Simple event ): ??? ????? ?????? ?? ???? ???? ??? ${\displaystyle \{1\}}$ ?? ????? ????? ??? ????? .

????? ?????? ( Compound event ): ????? ?????? ?? ???? ?? ???? ??? ${\displaystyle \{2,4,6\}}$ ??? ????? ???? ?? ????? ????? ??? ?????.

????? ???????? : ????? ???? ?? ???? ?? ???? ???? ???? ????? ${\displaystyle 7}$ ?? ????? ????? ??? ?????. ????? ??????: ????? ???? ??? ???? ????? ?????? ???? ???? ??? ??? ?? ${\displaystyle 7}$ ?? ????? ????? ??? ?????.

??????? ?????????? ( Mutually Exclusive events ): ??????? ?????? ?? ?????? ?? ?? ???? ???????? ???????? ??????? ?? ${\displaystyle A\cap B=F}$ ??? ${\displaystyle \{2\},\{3\}}$? ????? ???????? ??? ???????.

??????? ???????? (dependent events): ????????? ?? ??????????. ??? ????? ????? ??? ????? ??? ????? ????:

${\displaystyle P(1)=P(2)=P(3)=P(4)=P(5)=P(6)=1/6}$

??????? ??????? ( Exhaustive events ): ??? ??? ${\displaystyle S}$ ???? ???? ?? ??? ??????? ${\displaystyle A,B,C}$ ????? ??? ???? ?????? ??????? ??????:

1. ??????? ???? ????? ??: ${\displaystyle A\cap B=F}$ ? ${\displaystyle A\cap C=F}$ ? ${\displaystyle C\cap B=F}$
2. ???? ???? ???? ????? ?? : ${\displaystyle A\neq F}$ ? ${\displaystyle B\neq F}$ ? ${\displaystyle C\neq F}$
3. ??????? ????? ${\displaystyle S}$ ?? : ${\displaystyle A\cup B\cup C=S}$

??????? ??????? (Complementary events): ??????? ?????? ??????? ????? ???? ?????? ????? ${\displaystyle A}$ ??? ??? ${\displaystyle A'}$ ????? ?????? ??? ${\displaystyle A\cup A'=S}$

??????? ????????? ( Independent events ): ?????? ?? ????? ?? ???? ?????? (??? ????? ?? ???? ?? ????? ????? ?? ??? ???? ?????) .

????? ????? ?????????? ??????? ???????? ${\displaystyle P(A\cap B)=P(B)*P(A)}$ ???? ????? ??? ??????? ????? ?? ??? :

${\displaystyle P(A\cap B\cap C\cap ...\cap Z)=P(A)*P(B)*P(C)*...*P(Z)}$

??????? ????? ?????? (????????) Conditional Probability: ????? ???? ?????? ???? ?? ???? ????? ??? ??? ???? ?? ????? ????? ??? ????? ??? ???? ?????? ??? ???? ????? ???? ?????? ???? ??? ??????? (?? 52 ??? 51) ???????? ${\displaystyle A}$, ${\displaystyle B}$ ???? ??? ???? ${\displaystyle A}$ ???? ???? ${\displaystyle B}$ ??????? ${\displaystyle A/B}$ ?????:

${\displaystyle P(A\mid B)={\frac {P(A\cap B)}{P(B)}},}$

???? ?? ??????? ?? ????? ??? ????? ?????? ?? ????? ??? ???? ?? ????? ?? ????? .

${\displaystyle P(A/B)s}$ ??? ?????? ???? ????? A ???? ???? ????? B ? ?? ??? ????? ???? ???? ????? ?????? ????? ??? .

??? ???? ??????? ??????? ?? ?? ???? ???? ?????? ??? ????? ( when A and B are independent events ) ???? ???????:

${\displaystyle P(A\cap B)=P(B)*P(A)}$

????:

????? ???? 14 ??? ???? 8 ?????? 6 ????? ???? ????? (????????) ?? ??????? ??????? ???? ?????? ??? ????? ( ?? ??? ????? ???? ).

???? ?????? ?? ???? ??????? ????? ?????? (?????? ????? ???????? ?????). (???? ?????).

????:

???? A = ??? ??? ??? ????? ?????

????? B = ??? ??? ??? ????? ?????

???????? ?? ${\displaystyle P(A/B)s}$ ??? ${\displaystyle A}$ ?????? ???????? ${\displaystyle B}$ ?????? ??????.

${\displaystyle P(A\cap B)=P(B)*P(A)}$

${\displaystyle P(A\cap B)=(6/14)*(8/13)=24/91=0,2637}$

???? ??? ????? ??? ????? = ?(?? ?) + ?(?? ?) = (8÷14)×(7÷13) + (6÷14)×(5÷13) = 0.4725 ???? ??? ????? ???????? ?? ????? = ?(?? ?) + ?(?? ?) = 0.2637 + 0.2637 = 0.5274 ???? ????? ?????????? ???????? 0.4725 + 0.5274 = 0.9999 ? 1

????? ???????? [ ??? ]

1) ??? ??? ${\displaystyle A}$ ??? ?? ${\displaystyle S}$ ?? ???? ${\displaystyle A}$ ?????? ????? ?? ${\displaystyle S}$ ???: ${\displaystyle P(A)}$ ???? ?? ?????? ???? ????? ${\displaystyle A}$

?????? ???? ????? ${\displaystyle A}$: ????? ??? ????? ???? ????? ${\displaystyle A}$ ?????? ????? ??? ?? ??????? ???? ???? ?????? .

2) ??????? ?????????? (?????????) : ${\displaystyle A\cup A'=S}$ ??? ????: ${\displaystyle P(A)+P(A')=1}$ ????? ???????: ${\displaystyle P(A')=1-P(A)s}$ ?? ${\displaystyle P(A)=1-P(A')s}$

????? ???? ?? ????? ${\displaystyle A'}$?? ??? ??? ???? ${\displaystyle A}$ .

3) ????? ???????? ??????? ??????? ????? ?????? ?????? ??? ??????? ????? ${\displaystyle S}$

4)??????? ?????????? ${\displaystyle B}$, ${\displaystyle A}$ ?? ??????? ${\displaystyle F}$ ???: ${\displaystyle P(A\cap B)=0}$, ${\displaystyle P(A\cup B)=P(A)+P(B)}$ , ≪????? ????? ??? ??? ???? ?? ????? ????????≫.

5) ??? ??? ${\displaystyle A}$, ${\displaystyle B}$ ????? ??? ???????? (??????) ?? ?????? ???? ????? ??? ????? ???: ${\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)}$ ????? ????? ??? ???????? ${\displaystyle P(A\cap B)s}$ ??????? ????? ??? ???? ???????? ????? ??????? ??? ${\displaystyle A}$, ${\displaystyle B}$ ??? ???? ??? ?? ${\displaystyle A}$ ????? ?? ${\displaystyle B}$.

???? ????? ??????? ??????? ????? ?? ????? ?????? ???????:

${\displaystyle P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(A\cap B)-P(A\cap C)-P(B\cap C)}$

6) ??? ??????? ?? ???? ??????? ${\displaystyle S}$ ??????? ????????? ?? ${\displaystyle 2^{N}}$??? ${\displaystyle N}$ ??? ????? ?????? ${\displaystyle S}$ ???? ????? ????? ????? ??? ????? ??? ????? ?? ${\displaystyle 2^{6}=64}$ ????? ??? ???? ??????? ???????? ${\displaystyle \varnothing }$ ??????? ${\displaystyle S}$ ??? : ${\displaystyle S=\{1,2,3,4,5,6\}}$

???????:

(1) ?? ????? ????? ???? ???? ???? ????? ????? ????? ???? ???? ??????? ${\displaystyle S}$.

???? : ???? ?????? ??? ?????? ${\displaystyle H}$, ${\displaystyle T}$ ???? ??????? ???? ????? ?? ${\displaystyle 6}$ ????? ?? ?????? ?? ${\displaystyle 1}$ ??? ${\displaystyle 6}$ ????? ???? ??? ????? ???? ??????? ${\displaystyle 2*3=12}$ ??:

${\displaystyle S=\{(H;1);(H;2);(H;3);(H;4);(H;5);(H;6);(T;1);(T;2);(T;3);(T;4);(T;5);(T;6)\}}$

????? ??????? ???????? ??????? :

${\displaystyle S=\{(H;1);(H;2);...;(T;5);(T;6)\}}$

(2) ???? ??? ????? ??? ?? ??? ???? ${\displaystyle 10}$ ???? ??????? ?????? ??????? ${\displaystyle 3}$ ?????? ${\displaystyle 2}$ ?????? ${\displaystyle 5}$ ????? ??? ?????? ?? ???? ????? ???????? ?????

????  : ??? ?????? ???? ???? ??????? (????? ?????) ?? ${\displaystyle 3}$ ???? ?????? ???? ???? ?? ???? ????? ${\displaystyle 10}$???????? ?? ${\displaystyle A}$?? ??? ????? ????? ????? ???????: ${\displaystyle P(A)=M/N=3/10=0.3}$.

(3) ??? ??? ?????? ???? ??? ?? ${\displaystyle 0,05}$ ??? ?????? ?? ?????

????: ???? ?? ???????? ??????? ?? ????? ?????? ???????? ?????? ?? ?? ??????? ????? ?????? ?????? ????? ??:

${\displaystyle A}$ : ??? ?? ???? ????? ? ${\displaystyle A'}$ : ??? ?? ???? ????? ??? :

${\displaystyle P(A)=1-P(A')=1-0.05=0.95}$.

(4) ??? ?? ???? ??????? ?????? ????? (???? ??????) ??? ?????????? ${\displaystyle 0,1}$? ${\displaystyle 0,3}$? ${\displaystyle 0,6}$ ?? ????? ????? ??????? ???? ?????

???? : ??? ???? ????? ??? ?? ???? ??????? ????? ?????? ?????? ??????? ??? ??: ${\displaystyle 0.1+0.3+0.6=1}$ ???????? ?????.

(5) ??? ?? ???? ??????? ?????? ?????? ????? (???? ??????) ??? ?????????? ${\displaystyle 0.6,\ 0.3,\ 0.1,\ 0.0}$

???? : ??? ???? ????? ??? ?? ?? ???? ???? ???? ?? ????? ${\displaystyle F}$ ???? ???? ???????? ??????? ????? ???? ????? ${\displaystyle =}$ ${\displaystyle F}$ ???????? ??? ?????.

(6) ??? ??? ?????? ?????? ?? ???? ????????? ?? ${\displaystyle 0,45}$??????? ?????? ?? ???? ??????? ?? ${\displaystyle 0,65}$??????? ?????? ?? ???????? ???? ?? ${\displaystyle 0,37}$???? ?????? ?????? ?? ??? ???????? ??? ?????.

???? : ?????? ???? ?????????? ??????? ??????? ???? ????:

${\displaystyle A}$: ?????? ?????? ?? ???? ?????????

${\displaystyle B}$ : ?????? ?????? ?? ???? ???????

${\displaystyle A\cap B}$: ?????? ?????? ?? ???????? ???? ?????:

${\displaystyle P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.45+0.65-0.3=0.73}$.

???? ????? [ ??? ]

??????? [ ??? ]

????? ???????? ?? ???????? ??????? :
	??? ?????? ????? ?? ?????. ???? ?? ???? ???????.

• ????? ???????
• ????? ?????