[01~02] 01 3 5 1 6 6 3 3 6 2 5 5 1 2+1=3 04 7 2 3 5 7 4 7 7 1 7 7 4+1-1=4 02 1 6 6 1 3 5 3 2 4 6 3 3+3=6 [05~06] 05 2 3 a b (a b) a+b=2 (1 1) 1 a+b=3 (1 2) (2 1) 2 1+2=3 [03~04] 1 10 10 1 03 4 5 4 4 8 2 5 5 10 2 2+2=4 06 5 10 a b (a b) a+b=5 (1 4) (2 3) (3 2) (4 1) 4 a+b=10 (4 6) (5 5) (6 4) 3 4+3=7 160
[07~08] 07 2 4 6 8 4 1 3 5 7 9 5 4_5=20 [11~13] (a+b)(p+q+r)(x+y+z) 11 2_3_3=18 08 1 3 5 7 9 5 5_5=25 [09~10] 09 36 36=2 _3 36 2å_3 a=0 1 2 b=0 1 2 a 3 b 3 36 3_3=9 12 a a 3_3=9 10 13 p 200 p 200=2 _5 200 p 2å_5 a=0 1 2 3 b=0 1 2 a 4 p b 3 2_3=6 200 p 4_3=12 18-6=12 161
[01~03] 01 P 4! 4! P = = =4_3_2=24 (4-3)! 1! 05 3 5 3 P =5_4_3=60 02 P 6! 6! P = = =6_5_4_3_2_1=720 (6-6)! 0! 03 Pº 5! 5! Pº= = =1 (5-0)! 5! [06~07] 06 5 (5-1)!=4!=4_3_2_1=24 [04~05] 5 1 2 3 4 5 5 04 5 5 P =5!=5_4_3_2_1=120 07 4 2 2 3 2!=2 2 2!=2 2_2=4 162
[08~10] 08 P 5 =25 12 3 1 2 3 4 3 4 P =3 =81 09 P 3 =27 10 P 2 =16 [13~14] 5 1 1 2 2 3 13 5 5 2 1 2 2 5! =5_3_2=30 2!_2! [11~12] 11 2 a b 3 2 3 P =2 =8 14 5 1 2 1 2 2 3 3! =3 2! 163
[01~03] 01 C 3! (3-2)!_2! 3! = =3 2! [06~07] 10 06 2 10 2 10_9 ºC = =45 2 02 03 C 6! (6-2)!_2! C 5! (5-5)!_5! 6! = =15 4!_2! 5! = =1 5! 07 4 1 1 9 3 9_8_7 ªC = =84 3_2_1 [04~05] n 04 «C =28 n! n(n-1) = =28 (n-2)!_2! 2 n(n-1)=8_7 n=8 [08~09] 4 10 3 08 3 4 3 C =4 05 «C =35 n! n(n-1)(n-2) = =35 (n-3)!_3! 6 n(n-1)(n-2)=7_6_5 n=7 09 1 4 1 6 2 C _ C =4_ 6_5 2 =4_15=60 164
[10~12] 10 H C = C = C =4 [15~16] 4 15 4 3 4 3 6_5_4 H = C = C = =20 3_2_1 11 H 6_5 C = C = C = =15 2 16 4 5 12 H C =ªC =ªC = 9_8_7_6 4_3_2_1 =126 4 5 8_7_6 H = C = C = C = =56 3_2_1 [13~14] n 13 H =«C H = C = C n=5 [17~18] 17 x+y=5 H = C = C = C =6 14 «H = C «H =«C =«C = C n+2=7 n=5 18 x+y+z=7 9_8 H = C =ªC =ªC = =36 2 165
[01~04] 01 02 S(3, 2) 3 2 1 2 C =3 S(5, 4) 5 4 1 1 1 2 C =10 [05~06] 4 05 06 3 4 3 S(4 3) 3 2 1 4 2 S(4 3)= C =6 2 4 2 S(4 2) 2 2 1 3 C _ C =4 2 2 1 4_3 C _ C _ = =3 2! 2_2 S(4 2)=4+3=7 03 04 P(3, 2) 3 2 3=1+2 P(3 2)=1 P(5, 4) 5 4 5=2+1+1+1 P(5 4)=1 [07~08] 7 07 2 7 2 P(7 2) 7 2 7=6+1=5+2=4+3 P(7 2)=3 08 5 7 5 P(7 5) 7 5 7=3+1+1+1+1=2+2+1+1+1 P(7 5)=2 166
[09~12] 09 (a+b) Cºa + C a b+ C a b + C ab + C b =a +4a b+6a b +4ab +b [13~16] 13 (x-2)fl (x-2)fl C (-2)fl x r=0 Cº_(-2)fl =64 10 (x-2) Cºx + C x (-2)+ C x (-2) + C x(-2) + C (-2) =x -8x +24x -32x+16 14 (2x-3y)fi x y (2x-3y)fi C (2x)fi (-3y) = C 2fi (-3) xfi y r=1 x y C _2 _(-3)=5_16_(-3)=-240 11 {x+;[!;} fi 15 Cºxfi + C x {;[!;}+ C x {;[!;} + C x {;[!;} + C x{;[!;} + C {;[!;} fi 10 5 =xfi +5x +10x+ + + 1 x x xfi {x-;[@;} fi x {x-;[@;} fi C xfi {-;[@;} = C (-2) xfi 5-2r=1 r=2 x C _(-2) =40 12 {x-;]!;} fi 1 1 Cºxfi + C x {- }+ C x {- } 1 + C x {- } y y y 1 + C x{- } 1 + C {- } y y fi 5x 10x 10x 5x =xfi - + - + - 1 y y y y yfi 16 {x -;[#;} fl x {x -;[#;} fl C (x )fl {-;[#;} = C (-3) x 12-3r=3 r=3 x C _(-3) =20_(-27)=-540 167
[01~03] 01 6 6_6=36 6 (1 5) (2 4) (3 3) (4 2) (5 1) 5 05 2 3 5! 3 3!_2! 3!_2! =;1 0; 5! ;3 6; 02 03 1 6_6=36 1 (1 2) (2 1) (2 3) (3 2) (3 4) (4 3) (4 5) (5 4) (5 6) (6 5) 10 ;3!6);=;1 8; 6_6=36 3_3=9 ;3ª6;=;4!; [04~06] 2 3 04 3 2 3 5! 3 3 3! 3! 3!_3! 3!_3! =;1 0; 5! 06 07 08 2 3 5! C 3 3!_2! C _3!_2! C _3!_2! =;1 0; 5! [07~08] 5 A B C D E 3 A 5 3 C 3 A A C C =;1 0;=;5#; C C D 5 3 C 3 C D C D C C =;1 0; C 168
[09~10] 6 2 09 2 6 2 1 [13~15] 1 4 4 2 13 12 2 12 12 1 10 2 6 2 0 [11~12] 15 11 15 0 14 43 2 43 43 1 12 15 1 15 44 2 43 44 0 169
[01~02] A B P(A)=;4!; P(B)=;3@; 01 02 P(A;B)=;3!; P(A'B) P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=;4!;+;3@;-;3!;=;1 2; P(A'B)=;4#; P(A;B) P(A;B)=P(A)+P(B)-P(A'B) P(A;B)=;4!;+;3@;-;4#;=;6!; 04 6 6_6=36 6 A 12 B 6 (1 5) (2 4) (3 3) (4 2) (5 1) 5 P(A)=;3 6; 12 (6 6) 1 P(B)=;3 6; A;B=0 6 P(A'B)=P(A)+P(B)=;3 6;+;3 6;=;6!; [03~04] 03 5 3 6_6=36 5 A 3 B 5 (1 4) (2 3) (3 2) (4 1) 4 P(A)=;3 6;=;9!; 3 (1 4) (2 5) (3 6) (4 1) (5 2) (6 3) 6 P(B)=;3 6;=;6!; 5 3 (1 4) (4 1) 2 P(A;B)=;3 6;=;1 8; 5 3 P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=;9!;+;6!;-;1 8;=;9@; [05~06] 1 40 40 05 4 7 4 A 7 B 4 7 A;B 1 40 4 10 P(A)=;4!0);=;4!; 1 40 7 5 P(B)=;4 0;=;8!; 1 40 4 7 28 1 P(A;B)=;4 0; P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=;4!;+;8!;-;4 0;=;2 0; 170
06 5 9 5 A 9 B 5 9 A;B 1 40 5 8 P(A)=;4 0;=;5!; 1 40 9 4 P(B)=;4 0;=;1 0; 1 40 5 9 P(A;B)=0 P(A'B)=P(A)+P(B)=;5!;+;1 0;=;1 0; [07~08] 07 2 1 1 A AÇ 2 P(AÇ )=;2!;_;2!;=;4!; P(A)=1-P(AÇ )=1-;4!;=;4#; [09~10] 4 2 2 09 10 11 1 1 A AÇ 2 C P(AÇ )= =;1 5; C P(A)=1-P(AÇ )=1-;1 5;=;1!5$; 1 1 A AÇ 2 P(AÇ )= C C =;5@; P(A)=1-P(AÇ )=1-;5@;=;5#; [11~12] 4 3 3 1 1 A AÇ 3 C P(AÇ )= =;3 5; C P(A)=1-P(A C )=1-;3 5;=;3#5$; 08 3 1 1 A AÇ 3 P(AÇ )=;2!;_;2!;_;2!;=;8!; P(A)=1-P(AÇ )=1-;8!;=;8&; 12 1 1 A AÇ 3 C P(AÇ )= =;3 5; C P(A)=1-P(A C )=1-;3 5;=;3#5!; 171
[01~03] 1 10 10 01 3 A 3 B A={2 4 6 8 10} B={3 6 9} A;B={6} [04~05] A B P(A)=0.3 P(B)=0.5 P(A B)=0.4 04 P(A;B) P(A;B) P(A B)= P(B) P(A;B)=P(B)P(A B) =0.5_0.4=0.2 P(A)=;2!; P(A;B)=;1 0; A B P(A;B) P(B A)= =;1 0 ;2!;=;5!; P(A) 05 P(B A) P(A;B) 0.2 P(B A)= = =;3@; P(A) 0.3 02 03 A B A={2 4 6 8 10} B={2 3 5 7} A;B={2} P(A)=;2!; P(A;B)=;1 0; A B P(A;B) P(B A)= =;1 0 ;2!;=;5!; P(A) 5 A 5 B A={2 4 6 8 10} B={1 2 3 4 5} A;B={2 4} P(A)=;2!; P(A;B)=;5!; A B P(B A)= P(A;B) P(A) =;5! ;2!;=;5@; [06~08] A B P(A)=0.2 P(B)=0.6 P(A'B)=0.7 06 07 08 P(A;B) P(A;B)=P(A)+P(B)-P(A'B) =0.2+0.6-0.7=0.1 P(A B) P(A;B) 0.1 P(A B)= = =;6!; P(B) 0.6 P(A BÇ ) P(BÇ )=1-P(B)=1-0.6=0.4 P(A;BÇ )=P(A)-P(A;B) =0.2-0.1=0.1 P(A;BÇ ) 0.1 P(A BÇ )= = =;4!; P(BÇ ) 0.4 172
[09~11] A B 6 C Z Y 09 A B Y A={2 4 6} B={1 3 5} [12~14] 1 5 A 12 P(A) A={5 6} P(A)=;6@;=;3!; P(A)=;6#;=;2!; P(B)=;6#;=;2!; A;B=0 P(A;B)=0 P(A;B)+P(A)P(B) A B 13 4 A 2 10 A C Y A={2 4 6} C={1 2 3 6} P(A)=;6#;=;2!; P(C)=;6$;=;3@; A;C={2 6} P(A;C)=;6@;=;3!; A ;3!; A ;3@; C _{;3!;} _{;3@;} =;2 7; P(A;C)=P(A)P(C) A C 11 B C Z B={1 3 5} C={1 2 3 6} P(B)=;6#;=;2!; P(C)=;6$;=;3@; B;C={1 3} P(B;C)=;6@;=;3!; P(B;C)=P(B)P(C) B C 14 6 A 4 A ;3!; A ;3@; C _{;3!;} _{;3@;} fl =;2 4º3; 173
[01~03] X X 1 2 3 4 05 P(X=1) C _ C P(X=1)= =;5#; C P(X=x) ;8!; ;2!; ;4!; a 1 01 a 1 ;8!;+;2!;+;4!;+a=1 a=;8!; 06 P(X=2) C _ Cº P(X=2)= =;1 0; C 02 P(X=3) P(X=3)=;4!; 07 P(X=x)= 2C x _ 3 C 2-x 5C 2 (x=0 1 2) 03 P(1 X 2) P(1 X 2)=P(X=1)+P(X=2) =;8!;+;2!;=;8%; [04~07] 2 3 2 X [08~10] X X f(x) k 08 f(x)=k (0 X 3) y f(x)=k(0 x 3) x 1 3k=1 k=;3!; k O y=f(x) 3 x 04 P(X=0) 2 3 2 09 f(x)=kx (0 X 3) C = 5_4 2 =10 2 0 f(x)=kx(0 x 3) y 2 C =3 x 1 3k y=f(x) C P(X=0)= =;1 0; C ;2!;_3_3k=1 k=;9@; O 3 x 174
10 f(x)=k x (-1 X 1) y [13~15] X E(X)=10 V(X)=9 f(x)=k x (-1 x 1) x 1-1 ;2!;_1_k+;2!;_1_k=1 k=1 k O y=f(x) 1 x 13 Y=X+4 E(Y)=E(X+4)=E(X)+4=14 V(Y)=V(X+4)=V(X)=9 r(y)=r(x+4)=r(x)=3 [11~12] X X 11 X P(X=x) 0 1 2 ;4!; ;2!; ;4!; 1 E(X)=0_;4!;+1_;2!;+2_;4!;=1 V(X)=0 _;4!;+1 _;2!;+2 _;4!;-1 =;2!; r(x)= '2 2 14 Z=-X+1 E(Z)=E(-X+1)=-E(X)+1=-9 V(Z)=V(-X+1)=V(X)=9 r(z)=r(-x+1)=r(x)=3 12 X P(X=x) -1 0 1 ;5!; ;5@; ;5@; 1 E(X)=(-1)_;5!;+0_;5@;+1_;5@;=;5!; V(X)=(-1) _;5!;+0 _;5@;+1 _;5@;-{;5!;} =;2!5$; r(x)= '1 4 5 15 W=3X-10 E(W)=E(3X-10)=3E(X)-10=20 V(W)=V(3X-10)=9V(X)=81 r(w)=r(3x-10)=3r(x)=9 175
[01~04] X X 01 1 180 3 X B{180 ;6!;} E(X)=180_;6!;=30 V(X)=180_;6!;_;6%;=25 r(x)=5 04 10% 1000 X B{1000 ;1 0;} E(X)=1000_;1 0;=100 V(X)=1000_;1 0;_;1ª0;=90 r(x)=3'1 0 02 1 36 X B{36 ;2!;} E(X)=36_;2!;=18 [05~07] X E(X) r(x) X 05 E(X)=8 r(x)=1 X N(8 1) V(X)=36_;2!;_;2!;=9 r(x)=3 06 E(X)=20 r(x)=3 X N(20 3 ) 03 80% 250 X B{250 ;5$;} E(X)=250_;5$;=200 V(X)=250_;5$;_;5!;=40 r(x)=2'1 0 07 E(X)=30 r(x)=4 X N(30 4 ) 176
[08~10] X z N(50 10 ) 0.5 1.0 1.5 08 2.0 P(Xæ60) X-50 60-50 P(Xæ60)=P{ æ } 10 10 =P(Zæ1) =0.5-P(0 Z 1) =0.1587 P(0 Z z) 0.1915 0.3413 0.4332 0.4772 [11~12] 720 1 X z 0.5 1.0 1.5 2.0 P(0 Z z) 0.1915 0.3413 0.4332 0.4772 11 P(Xæ130) X B{720 ;6!;} E(X)=720_;6!;=120 V(X)=720_;6!;_;6%;=100 09 P(40 X 55) 40-50 X-50 55-50 P(40 X 55)=P{ } 10 10 10 =P(-1 Z 0.5) =P(0 Z 1)+P(0 Z 0.5) =0.3413+0.1915 =0.5328 X N(120 10 ) P(Xæ130) X-120 130-120 =P{ æ } 10 10 =P(Zæ1) =0.5-P(0 Z 1) =0.1587 10 P(30 X 65) 30-50 X-50 65-50 P(30 X 65)=P{ } 10 10 10 =P(-2 Z 1.5) =P(0 Z 2)+P(0 Z 1.5) =0.4772+0.4332 =0.9104 12 P(100 X 125) X N(120 10 ) P(100 X 125) 100-120 X-120 125-120 =P{ } 10 10 10 =P(-2 Z 0.5) =P(0 Z 2)+P(0 Z 0.5) =0.4772+0.1915 =0.6687 177
[01~04] m r n X 01 02 03 04 m=10 r=4 n=4 E(X )=m=10 r V(X )= =;; 4 ;=4 n r r(x )= =2 'n m=16 r=6 n=9 E(X )=m=16 r V(X )= =;; 9 ;=4 n r r(x )= =2 'n m=25 r=12 n=16 E(X )=m=25 r V(X )= =;; 1 6 ;;=9 n r r(x )= =3 'n m=300 r=35 n=49 E(X )=m=300 r 35 V(X )= = =25 n 49 r r(x )= =5 'n [05~07] N(50 10 ) z P(0 Z z) 25 X 0.5 1.0 1.5 2.0 0.1915 0.3413 0.4332 0.4772 05 06 07 P(X æ52) X E(X )=50 10 r(x )= =2 X ' 25 N(50 2 ) X -50 52-50 P(X æ52)=p{ æ } 2 2 P(X æ52)=p(zæ1) P(X æ52)=0.5-p(0 Z 1) =0.1587 P(46 X 53) X N(50 2 ) P(46 X 53) 46-50 X -50 53-50 =P{ } 2 2 2 =P(-2 Z 1.5) =P(0 Z 2)+P(0 Z 1.5) =0.4772+0.4332 =0.9104 P(48 X 54) X N(50 2 ) P(48 X 54) 48-50 X -50 54-50 =P{ } 2 2 2 =P(-1 Z 2) =P(0 Z 1)+P(0 Z 2) =0.3413+0.4772 =0.8185 178
[08~10] r n X m 95% 08 P( Z 1.96)=0.95 r=6 n=9 X =17 m 95% r r X -1.96_ m X +1.96_ 'n 'n 6 6 17-1.96_ m 17+1.96_ '9 '9 13.08 m 20.92 [11~13] 10 n p^ p 99% P( Z 2.58)=0.99 11 n=24, p^=0.6 p^-p n Z= 11112 p^(1-p^) æ 11123 n p 99% p^-2.58æ p^(1-p^) p p^+2.58æ p^(1-p^) 1111 1111 n n 0.6-2.58æ 0.6_0.4 p 0.6+2.58æ 0.6_0.4 1111 1111 24 24 0.342 p 0.858 09 r=20 n=16 X =60 m 95% 60-1.96_ 20 '1 6 50.2 m 69.8 m 60+1.96_ 20 '1 6 12 n=400, p^=0.8 p^-p n Z= 11112 p^(1-p^) æ 11123 n p 99% 0.8-2.58æ 0.8_0.2 p 0.8+2.58æ 0.8_0.2 1111 1111 400 400 0.7484 p 0.8516 10 r=40 n=100 X =120 m 95% 40 40 120-1.96_ m 120+1.96_ 'ƒ100 'ƒ100 112.16 m 127.84 13 n=2100, p^=0.7 p^-p n Z= 11112 p^(1-p^) æ 11123 n p 99% 0.7-2.58æ 0.7_0.3 p 0.7+2.58æ 0.7_0.3 1111 1111 2100 2100 0.6742 p 0.7258 179