Чепуренко В.А. Учебное пособие по курсу Теория вероятности

		ΩOO = {					}
						ΩP P = {
			} Ω = ΩOO+ΩP P
					On = {
n	Pn	= {		n			}
			}		5
				A =	n=2	(On + Pn)
2−n					P
	n		P (On) = P (Pn) = 2−n
O1, P1, O2, P2, ...					4
		5	1n + 1n =			1n =	15
		P (A) =
		n=2	2	2	n=1	2	16
		P			P
						B	D
		A
			C
							Ω
				P (AB) = P (A) P (B)
	A	B	AB C		¯ ¯	¯
					AB	C
P (BC) > P (B) P (C)				AB C A + B
					D = (BC) \ (AB)

BC = AB + D				A D
	C A + D	P (C) 6 P (A) + P (D)			P (BC)		>
					P (C)
	P (AB) + P (D)		P (AB)	= P (B)
>		>
	P (A) + P (D)		P (A)

P (A) > P (AB)

{	}	}	{
{		{	}
		}
+ P (A2) + P (A3) − 2		A1A2A3 A	P (A) > P (A1) +
p = P (A) q = P (B) r = P (A + B)
A · B	A
P (A B)	¯
	P AB

P

¯

¯

P

¯

¯

¯

B

P A + B

P A + B + AB

P

(A B) =

P

+

(A)

+ P (B) − 2 P (AB)

P (A B) = 2 P (A + B) − P (A) − P (B)

P A B¯ = 1 − P (A + B) + P (AB)

¯

¯

P (AC) P (BC)

P BC 6 P

B P (C)

P (ABC)

6

P2

C)

(P (A) = a

P (B) = b P (C) = c

P (ABC)

¯

¯

¯

¯

¯ ¯

P AC + BC

P AC + P BC

P ABC

P (AC

BC)

P

+ B)

\

C)

((A

A1, A2, A3

P (A1 + A2 + A3) = P (A1) + P (A2) + P (A3) − P (A1A2) −

−P (A1A3)−P (A2A3)+P (A1A2A3)

P (A1A2A3) = P (A1)+

+P (A2) + P (A3) − P (A1 + A2) − P (A1 + A3) − P (A2 + A3) +

+P (A1 + A2 + A3)

	P	n		n	Ai i = 1, ..., n A
		i=1 Ai		= i=1 P (Ai) −	16i<j6n P (Ai + Aj ) +
		Q		P	P	n Ai! .
+	P (Ai + Aj + Ak) − ... + (−1)n+1 P
1	X					X
	6i<j<k6n					i=1

¯

P BA

¯

P B + A

p = P (A) q = P (B) A B					P (A \ B)
	(A + AB + ABB		¯	¯
P (A B)		¯
		P B A	P A	B
P		+ . . .)
A1, A2, ..., An				Bm
			m

A1, A2, ..., An					k	P (Bm)	=	n−m		(−1)k Cmm+kSm+k
									P
					j=1 Aij !			k=0
Sk	=	16i1	<...<ik6n P
			P		Q	A, B, C
P (A B) 6 P (A C) + P (B C)
			A1, A2, ..., An
			P		n			P
					P
P (A1 A2 ... An) = i=1 P (Ai) − 2							16i<j6n P (AiAj ) +
1	P			(AiAj Ak) −		n+1 P			(A1	· ... · An) .
+ 4						... + (−2)

6i<j<k6n

P	(A \ B) + P (B) − (A + B) ;
	¯	¯	· B ;
P A + B		+ P A

P

¯ ¯ ¯ ¯

P A + AB + P A + AB ;

	¯
		B	− P A + B ;
P A
	AB +				−	¯ ¯
	¯	B				¯
P A			− P A B ;
	A + B +			¯
P		¯				P (A		B) ;
			BA
P				AB + AB ;

P

AC ABC¯

− P (ABC) ;

¯ ¯

¯

B ;

P AB + P (AB) − P A

P

((A + B) \

−

P

(A + B + AB) ;

A) + P (A)

P

(A B C) + P−A¯ B¯ C¯ ;

¯

P A + B + P (B) P (AB) ;

4 − ¯

A + B + (A + B

−

P

P

¯

P

¯

P

) + P (AB) ;

A¯ ·

+ P

(B)

− P

(B \ A) − P A¯ + B ;

P

A B + AB

P

(B + AB) ;

P A¯ · B + P A + B¯ + A · B¯

;

−

A +

·

−

A + B + [ (A) − (

¯

P

¯

P (A)

A B + P (B)

P B ;

A · B

P ¯

+ P

P

P B)] , (AB = B) ;

P

A¯ · B¯

(A \ B) + P B · A¯

+ P (ABBA) ;

P

¯

¯

(A +

−

A · B −

(A +

P

¯

AB + ABB + . . .) + P A ;

P

A + B + A B +

(AB) ;

¯

P

¯

¯

P

AB) ;

P A + B

P

A · B¯

+ P

·B · A¯ + P (AB) + P (A + B) ;

¯

¯

¯

¯

P

A · B

+ (AB)

·

+ ¯

·

(A +

−

¯ A + B +

¯

¯ A

B ;

P

¯

P

¯

¯

+ P B A + P A B ;

A + AB + A +

·

P

B)

P

¯

−

¯

P

¯

P

¯

(A) +

A ·

P

A + B +

(

P

B + A B + B ;

P

P

¯

B P (A + B + AB) ;

P

A + B¯

+ P

B + A¯

+ P

(B \ A) , (A B) ;

P

¯

¯

P

AB + ABA) ;

+

A · B

AB

+ (A) + (B) ;

(B \ A) +

(A) − (

P

¯

¯

¯

P

P

B + P

P

\

P

P A + B + AB) ;

¯

¯

−

−

A

¯

P A + B + P A

· B + P A + A + B¯

;

P

A¯ B¯

A + B + P

(AB) ;

P

·

P

((A B)

(AB))

P

(AC)

C ;

P A¯ B¯

+ (A B) + P A B¯

;

B · A

·

−

−

A +

+

A · B ;

P ¯

P

(AB) P (A) , (A B) ;

AB +

A · B − (A) +

(B) ;

P

¯

B

A

P (A) + P

¯

AB

P

¯

¯

¯

P

P

A + P

P

¯

·

(A + B)

−

P (ABC) .

ABC

Ω

|C|

P (A) =

Ω

n

n(A)

AB

lim n(A) =

n→∞ n

C C

|A| |Ω|,

A

lim n(B)

n→∞ n

1

36

A = {6♣}

m	n
{e1, e2, . . . , en}

E

E = {1, 2, 3}

m = 2

E

E

n

	Cnm =	n!
		m!(n−m)!

m

E

B = {9♠}

A C = {6} C

|Ω|

E =

m

m

m

n

m

m

	E
			n
m	Am =	n!
	n	(n−m)!

nm

E

E

			m
E
			m
E
	n		(n+m−1)!
Cm	= Cm	=
(n)	n+m−1		m!(n−1)!

n m

		m
	E
		m
	E
n	m	Am
		(n)
	n	m

m

= nm

n	n1	n2
(n1 + n2 = n)		m
m1	m2	(m1 + m2 = m)

Cm1 Cm2

P (n1, n2; m1, m2) = n1 mn2

Cn

	P (n1, n2; m1, m2) =	Anm11 Anm22 × Cmm1
		Am
		n

	Cmm1 n1m1 n2m2	m			n1
P (n1, n2; m1, m2) =		= Cm	1
	nm				n

.

	n2		.
m1		n	m2
	s (s		> 2)

	n	s	n1
n2	ns s (n = n1 + n2 + ... + ns)
	m
		m1
m2	ms s (m1 + m2 + ... + ms = m)

P (~n; m~) = P (n1, ..., ns; m1, ..., ms) =

					s
=		·	m	=	Cnmjj
	Cnm11				Q m .
			Cnm22 · ... · Cnmss		j=1
			Cn		Cn

m!

m1

ms

P (~n; m~) =

n1

·

... · ns

m1!m2!·...·ms!

=

= m1

! · ... · ms!

n1

·

nm

· ... ·

ns

!m2

n2

.

m!

n

m1

n

m2

n

ms

C= {

Ω= { 1 |Ω| = 4

P(C) =

}

}
}	P(C) = 1C = { }	|C| =
	4	Ω = {
1	}	|Ω| = 4
3		Ω = {
	|Ω| = 2		1
		P(C) =	2

A = {
} B = {			¯			}
AB A + B A \ B B \ A A B								A B

A

B

A

B

A |A| = 1 + 2 + ... + 5 = 15

B|B| = 12

	P(A) =	15	=	5	,	P(B) =	12	=	1	.
		36		12			36		3

AB

AB