# เอกสารประกอบการเรียน การวิเคราะห์ข้อมูลทางรัฐศาสตร์ บทที่ ๒

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216

1

2 2.1 P(A) = A 0 1 2.1.1 (Sample (Sample point)

space) S

1 1 1

1 1 2 3 6 = S : {1,2,3,4,5,6} 6 I, I = 1,2,,6 2 2

S : {, , , } 2

216

2

{0,1,2}

S : 0 2

1 1 1

2 2 2 (Event)

2.1.2

1 1 (Compound Event)

(Sample Event)

3 1 3

A 1 3 A : { 3} A : {4,5,6}

4 2 1 B : 1 2 B : {,,} 1-4

216

3

1. 1

(sample space) S = {1,2,3,4,5,6}

(event) A : 3 A : {4,5,6} C : 1 C : {,,}

2. S = {,, 2 ,}

(A) = A

P(A)

P (B) = 1 = 3/4 P (C) = 1 = 2/4

= P ( 3) = 3/6

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4

2.2 2.2.1 (Multiplications Rule) 2 1 = mn m 2 n

5 . . 4 . . . . 4 . . 5 1 2 3 4 5

. . 5 .

1 . 2 3 4 .

.

216

5

2.1 . . . 4x5 = 20

216

6

6 5 6 5x6 = 30 7 0,1,2,3,4,5,6,7,8,9 . 4 . . 7,000 . 2

4 4 10 ( 0-9) 10 ( 0-9) 10 ( 0-9) 9 ( 0) (. , . . ) 2.2.2 (Permutation) = 9x10x10x10 = 9,000

. 4

(Factorial) n! 1 n n! = n(n-1)(n-2)1

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7

: k n (k 0 P(A) > P(B)

= P(B) + P(AB )

= B (AB )

17 1 1 A B 3 P(AB) P(AB)

AB 3 3 A : {2,4,6} B : {1,2,3} S : {1,2,3,4,5,6} AB AB = {1,2,3,4,6} = {5}

P(AB) A B =1/6 P (AB) = P(A) + P(B) P(AB) = 3/6 + 3/6 - 1/6 = 5/6

P (AB) (1/6) + (1/6) = 5/6

= P(1) + P(2) + P(3) + P(4) + P(6) = (1/6) + (1/6) + (1/6) +

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(AND) (OR)1.

A B A B P(A or B) = P(A) + P(B) P(A and B) P (AB) P (AB) P(AB) P(AB) = = P(A) + P(B) P(AB) A B = P(A) + P(B)

2.

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

P(A and B) = P(A) + P(B) P(A or B) P(AB) =

A B

18 10

(A,B,C,D,E,F,G,H,I,J) 3 10 1. 3 10 2.

XX 2 XX 1

( 10 ) 3.

. XX 1

) 10!2)

3 10 = 10C3 = 3!(10-3)! = 1/120 = 120

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A : XX 1 2C1 = 2

1 2 XX = 2 8 = 8C2 = 28 8C2 = 2 x 28 = 56 P(A) = 56/1203)

XX 1 = 2C1 x = 0.4667

XX F G

B : XX 1 A : XX 1 C : XX 2

. P(A) = 0.4667

C : = 2C2 = 1 8C1 = 8

XX 2 2 XX 1 8 = P(C) = 1C1 x 8C1/ 10C3 = (1 x 8)/120 = 8/120 P(AC) = B P(B) = AC

AC = A C

= P(AC) = P(A) + P(C) P(AC) = 56/120 + 8/120 - 0

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= 0.4667 + 0.0666

= 0.5333

2.6

2 A : B : P(AB) = A

2

B B P(A B ) = A B A B

B A P(A B) > P(A B )

B 2.6.1 A B P(AB)

A B P(A B) P(A B) =

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23

2.6.2 A B P(A B) = P(A) P(A B) = P(B) A B P(A B) = P(AB) A B P(A) P(A B) = P(B) A B A B P(A B) = P(A)

P(B)

19 A B . P (AB) = 0.65, P(A) = 0.3, P(B) = 0.5 . P (AB) = 0.6, P(A) = 0.2, P(B) = 0.4

) P (AB) P(AB)

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

= 0.3 +

0.5 - 0.65 = 0.15 0 A B P(A B) = P(AB) / P(B) = 0.15/0.5 = 0.3 = P(A) P(A B) B A B =

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

P(A) A

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0.6 = 0

) P(AB)

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

= 0.2 + 0.4

A B P(A B) =

P(A)

P(AB) / P(B) = 0/0.4 = 0

A B 1. A B A B P(AB) = P(A) x P(B) P(A1, A2,An) = P(A1) xP(A2/A1)xP(A3/ A1,A2)P(Aa/Aa A1, A2,An P(A1, A2,An) = P(AB) = P(A) x P(B/A)

A B

2. A1, A2,An Aa-2A2A1)

1

P(A1)xP(A)xxP(An)

20 12 3 2 1 2 1

1. 2. 3.

1 G1 : i ; i = 1,2

) 2

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25

P(G1)

D1 ; i ; i = 1,2 = 8/11 = 3/11 = 9/12 P(D1) P(G2/ D1) P(D2/ D1) = 9/11 = 2/11 = 3/12

P(G2/ G1) P(D2/ G1)

G1 D2 D1 G2

D1 G2) D1)

P( 1 ) = P G1 D2) + P = P(G1)P(D2/ G1) + P(D1)P(G2/ = (9/12)(3/11) + (3/12)(3/11)

= (9/22)

2 3 1

9

1 =

P( 1 1 ) (9/22)

= 3C1 x 9C1 12C2

) 2 3 2

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26

9

0 = (1/22)

P( 2 ) = 3C2 9C0 ) 1 3 9

12C2

1 2 0 1

) + P( 2 ) 1

P( 1 ) = P( 1 =

P( 0 ) + P( 1 ) + P( 2 ) P( 1 ) + P( 2 ) = 1

= 1 P( 0 ) 3C0 9C2 12C2

= 1- 6/11 = 5/11

A B 1. A B 2. A B

3. A B 1. P(AB ) = P(A)P(B )

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27

P(B / A)

A B P(B/A) P(B)] P(AB )

P(AB )

P(A)

= P(AB ) = P(A)P(B / A)

= P(A)[1-P(B/A) = P(A)[1-

= P(B)

2. P(AB )

= P(A)P(B ) = P(B)P(A /B) = P(B)P(A )

= P(B)[1-P(A/B)] A

= P(B)[1-P(A)]

3. P(A B ) A

B

B A

= P(A )[1-P(B/ A )] = P(A )[1-P(B)] = P(A ) P(B )

= P(A )P(B / A )

n (C1, C2, ,Cn) S

B

U Ci = S CiCj = S

; i j = 1,2,,n A

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A P(Ci/A) =

P(A)

0 Cj

P(A/Ci)P(Ci)

P(A/Ci)P(Ci)

C1, C2, ,Cn (CiA)(CjA) = ; i j

P(A) = P(C1A) + P(C2A) ++P(CnA) P(A) = P(A/Ci)P(Ci) P(Cj/A) =

A = (C1A) (C2A) (CnA)

P(A/C1)P(C1) + P(A/C2)P(C2) ++ P(A/Cn)P(Cn) = P(C1A)

P(Cj/A) =

P(A/Cj)P(Cj) ..(1) P(A/Ci)P(Ci)

P(A)

Cj A (prior probability) P(Cj/A) (posterior probability)

(1)

(P(Cj/A)) P(Cj) P(Cj)

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29

21 . 5 30% . 20% . 2 45% 2 2 1. .

A1 : .

2. A2 :

R : 2 R :

2

A1 R A2

P(A1) P(R/ A1) P(R / A1) P(R /A2)

= 0.3 = 0.20

P(A2) = 1-P(A1) = 0.7 P(R/A2) = 0.45 = 1-0.2 = 0.8

= 1- P(R/ A1) = 1- P(R/A2)

= 1-0.45 = 0.55

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30

(1) A1 A2

(2) P(A) 0.3 0.7

P(R/Ai) P(Ai)P(R/ Ai) = P(Ai R) 0.06 0.2 0.45 (0.2)(0.3) = (0.7)(0.45) = 0.315 P(R) = 0.375

(3)

(4)

P(Ai /R) = (4)/(4) 0.16 0.06/0.375 = 0.315/0.375 = 0.84 1.0

(5)

1)

P( ./ 2 = P(A1/R) P(A1/R) = P(A1/R) P(R) R = A1 RA2R

A2R

P(R) = P(A1 R) + P(A2R) A1 R P(R/ A1) = P(A1 R)

P(Ai) P(A2)

P(A1 R) = P(R)

P(A1 R) = P(R/ A2) P(A2)

P(R/ A1) P(A1) + P(R/ A2)

= P(R/ A1) P(A1)

0.06 + 0.315 = 0.375

= (0.2)(0.3) + (0.7)(0.45) = = 0.06/0.375 = 0.16

P(A/R)

216 2)

31

P(/ 2 ) = P(A2/R) P(A2/R) = P(A2R) = P(R/ A1) P(A1) P(R)

(0.45)(0.7) = 0.84 0.375

=

0.375

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(Random Variable) 1. (Random Variable) () 1.1 (Discrete Random Variable)

2.

X X X X X X 145-180 145 X 180 3. (Probability Distribution) 1.2 (Continuous Random Variable)

X X S 3.1

2 X X

X1, X2,,Xn X = x X P(x) = P[X=x] 1. 2.

0 P(X=x) = P(x) 1

P(X=x) = 1 P(x) = 1

216 3.

33

P(A) = P(X=x) A S

= S =

1 4

{0,1,2,3,4} 30 X () = 0 1 8 8/30 = 0.27 2 4 4/30 = 0.13 3 4 4/30 = 0.13 12 12/30 = 0.4

4 2

2/30 = 0.07

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P(2) = P(X = 2) = 0.13 P(0) = P(X = 0) = 0.4 P(3) = P(X = 3) = 0.132.

P(1) = P(X = 1) = 0.27 P(4) = P(X = 4) = 0.07

1. 2

1 . A : 2 A : {2,3,4}

0.13+0.13+0.07 = 0.33

P(A) = P(X = 2) + P(X = 3) + P(X = 4)

=

. B : 1 B = {1,2,3,4}

4)

= 1- P(X = 0) = 1-0.4 = 0.6 3.2

P(B) = P(X = 1) + P(X = 2) + P(X = 3) + P(X =

X X

X

X X [a,b], a b X [a,b] P[X = a] = 0 X

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X P[X