Чепуренко В.А. Учебное пособие по курсу Теория вероятности

P

(A) = p ·

1 −

k(1−p)

< p

k 6= 0

p

D¯

n−kp

p

¯

P(B)

P ¯ (B) =

P(BD)

=

=

p

P ¯ (B) >

¯

P

D

1−

(D)

n−kp

D

> np

P(D)

k 6= 0

p 6= 0

k = n − 1

PD¯ (B) =

p

q = 1 − p

nq+p

P(C)

p

P ¯

(C) =

=

1−kp

1−P(D)

D

1

n

−

A = {

}

B = {

}

A = T1T2T3T4

Ti = {i

}2 1

4

3

P (A) = P (T1)·PT1 (T2)·PT1T2 (T3)·PT1T2T3 (T4) =

·

·

·

≈

36

35

34

33

≈ 1.7E − 05

B = K1T2T3T4 + T1K2T3T4 + T1T2K3T4 + T1T2T3K4

Ki = {i

}

4

4

P (B) = P (K1)

PK1 (T2)

PK1T2 (T3)

PK1T2T3 (T4) + ... =

·

36 · 35 ·

3

2

4

4

·3 2

4 ·

3 4 2

4

3 2 4

·

·

+

·

·

·

+

·

·

·

+

·

·

·

≈ 2.7E − 04

34

33

36

35

34

33

36

35

34

33

36

35

34

33

¯

¯

A

B

A

B

p = P (A) =

1

P (B)

PA (B) P ¯ (B)

(A) P ¯ (A)

2

PB

A

B

A

B

(ξ1, ξ2)

{(x1, x2) : x1, x2 [0; 1]}

r

Ar = {|ξ1 − ξ2| > r} Br = {ξ1 + ξ2 6 3r}

Cr = {max {ξ1

; ξ2} 612r} Dr

= {min

1{ξ1; ξ2} > r}

C3 = ξ1

2

2

2

> 0

1

,

2

−

C1

=

ξ1

6 2

C2 = ξ2

6 2

1

ξ

−

1

C1, C2, C3

ξ

ξ

C1, C2, C3

5%

8.9%

PB (A) > P (A)

PB (A) >

> P ¯ (A)

B

P (A) = p P (B) = 1 − ε

ε

PB (A) PA (B)

q (q > p)

PA+B · C¯

PA B (C)

A·B

¯ ¯

¯

P ¯ ¯ (C)

P ¯

¯

P ¯

P

C + B

C + B

A+B

C

A B

A

t = 0

t

(t; t + h)

λh + o(h), h → 0

Ak = {k

N

} Bk = {k

} Ci,k = {i

}

k

Ak

A1A2 A1A2

A1A2 + A1A2

Bk

Ci,k

A1B3C2,2

PA1 (A2)

PA2 (A1)

PA1

(A3)

PA1 (A3)

A2

PA2 (A3)

PA2 (B2)

PB1 (A1)

PA1A2 (C1,3)

PA1A2

(C2,1)

A3

k

} B = {

A = {

♦

}

k = 52 k = 54

Ak = {

k} k = 2, 3, 5

{A2, A3, A5}

D

P (A) = p P (B) = q

A

B

p

q

D

p

q

D

A \ B

A + B

A \ B

A + B

A B

A · (A + B)

A \ B

A + B

A + AB + B

A + B

A B

A \ B

A \ (A + B)

B

·A

·

AB

B \ A

A + B

A

(AB)

A

AB

B + A + AC

B ·

¯

A + B

\

A B

A + B + BA

A \ B

A B

AB + ABB

A · (AB + B)

AB \ B

A + B + AB

A + AC + B

A B B

A

A \ AB

AB A + AB

(B

\

A) + A

+ B + BC

A \ (AB)

A AB

A +

(A + B) (AB)

B

B + A + AC

B · A + B

\

AB

[a; b]

ξ η

C

D

E

[a; b]

A

B

C

D

E

ξ > 2η

ξ + η < 2

A + B

B

A

ξ < 3η

ξ + η > 3

A + B

B

A

ξ > 1

ξ + 2η < 3

A · B

B

A

2ξ > η

3 < 2η

B + A

A

B

ξ > 3η

2η − 2ξ > 1

A + B

B

A

ξ < 2η

ξ + η > 0

A + B

B

A

2ξ > η

ξ + η < 0

A B

B

A

2ξ > 3η

3 − ξ < η

B + A

A

B

2ξ > η

ξ + η < 1

A + B

A

B

ξ < 3η

ξ + 2η > 4

A + B

B

A

ξ > 2

2ξ + 2η < 5

A B

B

A

2ξ > 3η

3 + ξ < 2η

B + A

A

B

4ξ > 3η

3η − 2ξ > 1

A + B

B

A

ξ < η

3ξ + 2η > 2

A B

B

A

ξ > η

ξ + 2η < 4

A \ B

B

A

3ξ > η

3 − 2ξ < η

B + A

A

B

ξ > 2η

2ξ + 2η < 3

A \ B

A

B

2ξ > η

3 − ξ < η

B + A

A

B

ξ > 2η

ξ + 2η < 4

A \ B

A

B

2ξ > η

3 − 2ξ < η

B · A

B

A

ξ > 3η

ξ + 2η < 7

A + B

A

B

ξ < η

ξ + η > 2

A + B

A

B

ξ > η

ξ + 2η < 1

A + B

B

A

2ξ > 3η

5 < 2η + ξ

B + A

A

B

ξ > 2η

2η + ξ > 3

A + B

B

A

ξ < 3η

ξ + η > 2

A + B

B

A

2ξ > 5η

ξ + η < 2

A B

B

A

2ξ > η

4 − 2ξ < η

B + A

A

B

2ξ > 3η

ξ + η < 3

A + B

A

B

ξ < 3η

ξ + 2η > 6

A + B

B

A

ξ > 1

2ξ + 2η < 3

A + B

B

A

2ξ > η

3 + ξ < 4η

B + A

A

B

4ξ > η

3η + 2ξ > 5

A + B

B

A

ξ < 2η

3ξ + 2η > 4

A B

B

A

ξ > 2η

ξ + 2η < 6

A \ B

B

A

ξ > 2η

5 − 2ξ < η

B + A

A

B

ξ > η

2ξ + η < 5

A \ B

A

B

ξ > 3η

5 − 2ξ < η

B + A

A

B

ξ > 3η

3ξ + 2η < 4

A \ B

A

B

2ξ > 3η

7 − 2ξ < η

B · A

B

A

H1, H2, ..., Hn

Ω

A

n

X

P (A) =

P (Hi) PHi (A) .

i=1

H1, H2, ..., Hn

Ω

P

(A) P

(H

)

P

(A) P (H

)

PA (Hi) =

Hi

i

=

Hi

i

.

P (A)

n

P

P

(Hi) PHi (A)

i=1

N

N

{

C7

7

{

N

}

C7

N N

C7

H1 =

;

H2 =

;

H3 =

;

2

0

1

1

0

2

P (H1) =

C3 C4

=

1 P (H2) =

C3 C

4

= 4

P (H3) =

C3 C

4

= 2

2

2

2

7

A {

7

}

7

PH1 (A) =

10

6

5

PH2 (A) =

=

P

(A) =

10

10

H3

AB = PC (A + B) =

PC (A) + PC (B)

PC (A + B) =

P(AC+BC)

=

P(C)

P(AC)+P(BC) = PC (A) + PC (B)

PA (H1 + H3) =

P(C)

= PA (H1) + PA (H3) =

PH1 (A) P(H1)+PH3

(A) P(H3)

=

17

P(A)

41

} H2 = {

H1 = {

}

H3 = {

}

A1= {

}

3

P (H1) =

= 9

H1

33

P (H2)

3

3

PH1 (A) = 1−(1 − p)3 = p 3 − 3p + p2

P (H2) = h2!1!0!

2

1

3

i

× A3 =

3

H2

3!