1. 2. ;10!0; 16 01 02 154
03 04 05 155
06 110~116 P(A) 117~120 5~9 121 10 122~126 P(B A) 11~14 127~131 15~18 132 19 133~134 135 1~4 20~21 07 11~12/21 123~125 P(B A) 2 156
08 16 CardanoG 15011576 100 17 PascalB16231662 FermatP16011665 1655 HuygensC16291695 Bernoulli J 16541705 de Moivre A 16671754 18 19 LaplacePS17491827 20 HilbertD18621943 KolmogorovAN1903 1657 a p ap 1812 S AB P(S)=1P(u)=0 A 0 P(A) 1 AB P(A'B)=P(A)+P(B) 1999 1997 157
1 1 ;6!; 1 2 3 4 5 6 2 3 4 8 6 7 30 9 6 10 8 10 7 50 16 13 20 17 19 15 100 1 2 3 4 5 6 0.07 0.10 0.13 0.27 0.20 0.23 1 0.18 0.12 0.20 0.16 0.20 0.14 1 0.16 0.13 0.20 0.17 0.19 0.15 1 ;6!;(?0.17) (Stewart Culin 18581929) 2003 30 50 100 12 3456 1 2 3 4 5 6 30 50 100 123456 {1357} {37} {134} {1357} u 4 3 ;7#; U={123456} A={345}B={1356} A;B A'B AÇ A-B {35} {13456} {126} {4} 1 5 5 2 20 ;5$; 158
1 1 S A 3 B A;B A'B 1 S A 1 2 4 3 S={123456}S 5 6 A={235} A A A P(A) 1 H T 2 S A 3 1 S S={123456} A3 B A;B={3} A={235}B={36} A'B={2356} 2 S A 5 3 B 6 4 1 1 88 1 S={(HH)(HT)(TH)(TT)} A={(HT)(TH)} 159
2 AA AÇ S AB A B S A B A;AÇ=u A'B AB A;B S A B AB A;B=u S A B A B A B5 C A={246}B={135}C={5} 2 p.107 A A AÇ S A AÇ AB A;AÇ =u A AÇ A;B=u AB BA A'AÇ S AC A;C=uA C CA A4 B A={135} B={4} S A 2 1 3 5 6 B 4 A'B={1345}AÇ ={246}A;B=u A B 2 2 A B C ABC 88 290 1 H T 2 S S={(HH)(HT)(TH)(TT)} A A={(HH)(HT)(TH)} B B={(HT)(TH)} C C={(HH)} A;B={(HT)(TH)} B;C=u C;A={(HH)} B C AÇ={(TT)} BÇ={(HH)(TT)} CÇ={(HT)(TH)(TT)} ABC H T 1 S S={HT} 2 2 S S={(HH)(HT)(TH)(TT)} 2 =4 3 S S={(HHH)(HHT)(HTH) (THH)(HTT)(THT) (TTH)(TTT)} 2 =8 n S 2«160
3 89 390 2 2005 10575 1517 1864 1864?0.176 10575 3 3 n (n) (r«) r«{ } n 100 300 500 800 1000 46 158 241 403 502 0.460 0.527 0.482 0.504 0.502 n 0.5 0.5 n A r«r«n p n p A 2000 36 36 =0.018 2000 r«n p n 611 1214 1517 1821 621 2000 4073 1869 2166 3275 11383 2005 4016 2058 1864 2637 10575 www.nso.go.kr 2005 1517 r P(A) P(A)= ;4!; pr = pr =;4!; 161
4 P(A) P probability 123456 ;6!; A P(A) n A r r A n r P(A)= n 1 6 2 6_6=36 (x y) 6 A A={(1 5) (2 4) (3 3) (4 2) (5 1)} P(A)=;3 6; yy` 2 B B={(1 3) (2 4) (3 5) (4 6) (6 4) (5 3) (4 2) (3 1)} P(B)=;3 6;=;9@; yy` n A r P(A) P(A)= r n 5 2 2 4 3 5 4 2 2 4 A {1 3 5} A P(A) P(A)=;6#;=;2!; 3 8 4 C 3 2 5 2 C _ C C _ C = 3_10 =;7#; yy` C 70 4 (Laplace, P. S. ; 1749~1827) n r A n A r A P(A) P(A)= r n 89 4 2 =8 2 ;8@;=;4!; 6_6=36 4 A A={(13)(22)(31)(26)(35)(44) (53)(62)(66)} P(A)=;3ª6;=;4!; B B={(11)(22)(33)(44)(55)(66)} P(B)=;3 6;=;6!; 162
C =;12$0;=;3 0; ºC C _ C =;1 2 0;=;1 0; ºC 90 3 6 10 4 3 3 2 1 ;1 0 0º0;=;1 0 0; 1 1000 410 2 4 2 2 2 4 2 6!=720 2 5!_2=240 5!_2 240 = =;3!; 6! 720 1 1 7 7 2 2 3 3 1 1 ;7@; 6 3!_3!_2=72 1 1 4 1 1 5 2!_2!_2_5=40 ;7$2);=;9%; Æ æ 17 2 24 12 1 2 24 12 {;3#6%;}2 4 =0.509y 50.9 % Laplace, P. S. 17491827 18 (dalembert, J. L. R.; 1717~1783) 1773 Traite de mecanique celeste517981827 Theorie analytique des probailites 1812 163
5 2 1 2 n(a) A 1 10 1 10 8 6 5 7 9 3 2 4 2 3 3 5 S S A u,a,s 0 n(a) n(s) n(a) 0 P(A)= 1 n(s) n(s) n(u) P(S)= =1P(u)= =0 n(s) n(s) AB n(a'b)=n(a)+n(b)-n(a;b) 2 A3 B5 C 1 A;B 2 3 6 n(a;b)=1 n(a)=5n(b)=3 n(a'b)=n(a)+n(b)-n(a;b) =5+3-1=7 1 7 2 B;C=u n(b;c)=0 n(b)=3n(c)=2 n(b'c)=n(b)+n(c)-n(b;c) =3+2-0=5 2 5 P(A)=1 A P(A)=0 A A 0 P(A) 1 S P(S)=1 (u) P(u)=0 5 SA 0 P(A) 1 u,a,s 0 n(a) n(s) n(a) 0 P(A)= 1 n(s) A 0 1 A=S P(A)=1 A 1 n(s) P(S)= =1 n(s) A=u P(A)=0 A 0 n(u) P(u)= =0 n(s) 164
6 S AB AB A;B+u P(A'B)=P(A)+P(B)-P(A;B) AB A;B=u P(A'B)=P(A)+P(B) 6 S AB n(a'b)=n(a)+n(b)-n(a;b) A'B n(a'b) n(a) n(b) P(A'B)= = + - n(s) n(s) n(s) P(A'B)=P(A)+P(B)-P(A;B) AB A;B=u P(A;B)=0 P(A'B)=P(A)+P(B) n(a;b) n(s) AB P(A'B)=P(A)+P(B)-P(A;B) AB A;B=u P(A'B)=P(A)+P(B) 1 1 30 30 3 4 5 7 SABC n(a'b'c) =n(a)+n(b)+n(c)-n(a;b)-n(b;c) -n(c;a)+n(a;b;c) P(A'B'C) =P(A)+P(B)+P(C)-P(A;B)-P(B;C) -P(C;A)+P(A;B;C) ABC A;B=uB;C=uC;A=u A;B;C=u P(A;B)=0P(B;C)=0P(C;A)=0 P(A;B;C)=0 P(A'B'C)=P(A)+P(B)+P(C) S n A A ya n A i ;A j =u(i+j, i, j=1, 2, y, n) P(A 'A 'y'a n )=P(A )+P(A )+y+p(a n ) 3 A4 B A;B 12 A={3 6 9 y 30}B={4 8 12 y 28}A;B={12 24} P(A)=;3!0);P(B)=;3 0;P(A;B)=;3 0; 93 1 94 1 2 2 3 2 A C _ C P(A)= =;5#6);=;2!8%; C 3 B Cº_ C P(B)= =;5!6);=;2 8; C A B P(A;B)=0 2 A'B P(A'B) P(A'B)=P(A)+P(B) =;2!8%;+;2 8; =;7%; 165
7 S A A A C P(AÇ ) A;AÇ =u S A AÇ AÇ A A'AÇ =S P(S)=P(A'AÇ ) P(S)=P(A)+P(AÇ )=1 P(AÇ )=1-P(A) n(aç )=n(s)-n(a) n(aç ) n(s)-n(a) P(AÇ )= = n(s) n(s) =1-P(A) 7 1 P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=;3!0);+;3 0;-;3 0;=;3!0%;=;2!; yy` 5 C7 D P(C)=;3 0;P(D)=;3 0; C D P(C;D)=0 P(C'D)=P(C)+P(D)=;3 0;+;3 0;=;3!; yy` 3 5 3 2 A A AÇ A AÇ S A AÇ A'AÇ =S P(S)=P(A'AÇ )=P(A)+P(AÇ )=1 P(AÇ )=1-P(A) A AÇ P(AÇ )=1-P(A) 3 A P(A)=;6@;=;3!; 3 93 5 2 A 1 A AÇ C _ Cº P(A)= =;2!8);=;1 4; C P(AÇ )=1-P(A) 93 3 94 3 2 A A AÇ P(A)=;3 6;=;6!; P(AÇ )=1-P(A) =1-;6!;=;6%; =1-;1 4;=;1ª4; P(AÇ )=1-P(A)=1-;3!;=;3@; A F A F A F B E B E B E C D C D C D A A A C C =20 2 P(A)=;2 0;=;1 0; P(A C )=1-P(A) =1-;1 0;=;1ª0; 166
1 2 A3 B5 C 2 3 A'B A;B 6 P(A)=;2!;P(B)=;3!;P(A;B)=;6!; P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=;2!;+;3!;-;6!;=;6$;=;3@; 35 B'C P(B)=;3!;P(C)=;6!; B C P(B;C)=0 P(B'C)=P(B)+P(C) =;3!;+;6!;=;2!; 2 2 2 6 4 2 2 A A AÇ C P(A)= =;1 5; ºC P(AÇ )=1-P(A)=1-;1 5;=;1!5#; yy` 3 5 3 2 1 4 1 A F B E C D 1 2 3 3 5 2 5 2 A A AÇ 4!_2 P(A)= =;5@; 5! P(AÇ )=1-P(A) =1-;5@;=;5#; 1 50 1 12 53 10 1 2 ;2 5; 2 1 10 10 2 3 3 A 3 A A C 3 012 BºB B Bº={369}B ={14710}B ={258} 3 Bº B B C C _ C P(A)= + ºC ºC 3+12 15 = = =;3!; 45 45 P(AÇ )=1-P(A) =1-;3!;=;3@; 167
1 3!_2 =;2!; 4! 32 jk 2! 34 jk 2! 4 jk 3! 2!+2!+3! =;2!4);=;1 2; 4! 2 4 (1 5)(2 6)(6 2)(5 1) jk 4 5 (1 6)(6 1) jk 2 ;3 6;=;6!; 6 (1 6)(2 3)(3 2)(6 1) jk 4 12 (2 6)(3 4)(4 3)(6 2) jk 4 18 (3 6)(6 3) jk 2 24 (4 6)(6 4) jk 2 30 (5 6)(6 5) jk 2 36 (6 6) jk 1 6 4+4+2+2+2+1=15 ;3!6%;=;1 2; 5 (1 4)(2 3)(3 2)(4 1) jk 4 10 (4 6)(5 5)(6 4) jk 3 ;3 6; 3 ax-b=01 a=b (1 1)(2 2)(3 3)(4 4)(5 5)(6 6) jk 6 ax-b=02 2a=b (1 2)(2 4)(3 6) jk 3 6+3 =;3ª6;=;4!; 36 4 12 (57) C _ C =792 ªC _ C =36 ªC _ C =126-1 1 1234 3200 2 4 6 5 10 3 AB ab ax-b=0 1 2 4 12 5 7 12 5 10 4 3 36+126 =;7!9^2@;=;4ª4; 792 5 C 1- =1-;1 2º0;=;6%; ºC 168
2 2 3 2004 P P =;1 3º5 1ª7;?0.593 2005 P 2004 P =;1 4 5 5 2;?0.587 2006 P P =;1ª5 3ª4 0;?0.638 2004 2005 59%2006 64% 320042006 2004 2005 2006 8019 8544 9791 26354 5498 6008 5549 17055 13517 14552 15340 43409 wwwkmdborkr 3 A={x x 20} B={x x 20 } n(a) n(b) n(a;b) 6 9 3 AB 3 ;9!; 40 1 B O 251 ;1 8; A B O AB 12 9 14 5 ;4@0#; 169
1 DVD 1 2 200 100 300 550 150 700 750 250 1000 DVD 1 P(B A) 1 P 300 3 P = = 1000 10 P 200 2 P = = 1000 10 P 200 P = = 2 300 3 ;1 0º0º0; 200 2 P = = = P 300 P ;1 0º0º0; 1 S A B n(s)=1000n(a)=300n(a;b)=200 n(a;b) n(a) =;3@0)0);=;3@; 1 S AB n(s)=a+b+c+d n(a)=a+b n(b)=b+c S A a b B c d n(a;b)=b(n(a)+0n(b)+0) A A A B A B b b a+b+c+d P(B A)= = a+b a+b a+b+c+d P(B A)= P(A;B) P(A) 170
A B A BÇ a a a+b+c+d P(A;BÇ ) P(BÇ A)= = = a+b a+b P(A) a+b+c+d S A B S AB A B A B P(B A) n(a;b) n(a;b) n(s) P(A;B) P(B A)= = = n(a) n(a) P(A) n(s) A B A B c c a+b+c+d P(AÇ;B) P(B AÇ )= = = c+d c+d P(AÇ ) a+b+c+d A B A BÇ d d a+b+c+d P(BÇ AÇ )= = c+d c+d a+b+c+d P(BÇ AÇ )= B A B AÇ B A B AÇ P(A B)= P(AÇ B)= P(A BÇ )= P(AÇ BÇ )= P(A;B) P(B) P(AÇ;B) P(B) P(A;BÇ ) P(BÇ ) S AB P(A)>0 P(A;B) 0 P(B A)= 1 P(A) P(A;S) P(A) P(S A)= = =1 P(A) P(A) P(A;u) P(u) P(u A)= = =0 P(A) P(A) P(AÇ ;BÇ ) P(BÇ ) P(AÇ;BÇ ) P(AÇ ) 1 A B P(B A)= P(A;B) P(A) 200 P(A;B) 1000 P(B A)= = =;3@; P(A) 300 1000 100 P(A;BÇ ) 1000 P(BÇ A)= = =;3!; P(A) 300 1000 P(AÇ ;B) P(B AÇ )= =;1!4!; P(AÇ ) P(AÇ ;BÇ ) P(BÇ AÇ )= =;1 4; P(AÇ ) (P(A)>0) B BÇ A 200 100 300 AÇ 550 150 700 750 250 1000 AB P(A)=0.5P(A;B)=0.2 P(B A) n B B B yb«b ;B ;y;b«=u P(B 'B 'y'b«a) P(A;(B 'B 'y'b«)) = P(A) = = P((A;B )'(A;B )'y'(a;b«)) P(A) P(A;B )+P(A;B )+y+p(a;b«) P(A) P(A;B ) P(A;B ) = + +y+ P(A) P(A) =P(B A)+P(B A)+y+P(B«A) P(A)>0 P(A;B) P(B A)= P(A) P(B A)= 0.2 0.5 =;5@; P(A;B«) P(A) 100 1 171
101 1 A B A AÇ B 50% 10% 60% BÇ 20% 20% 40% 70% 30% 100% 1 50 % 30 % A B P(A)=;1 0º0;=;2!;P(A;B)=;1 0º0;=;1 0; ;1 0; P(A;B) P(B A)= = =;5#; yy` P(A) ;2!; P(A;BÇ ) 0.2 P(BÇ A)= = =;7@; P(A) 0.7 2 70 % 60 % 50 % 2 A B P(B A)= P(A) P(A;B)=P(A)P(B A) B A P(A B)= P(B) P(B;A)=P(B)P(A B) P(A;B)=P(B;A) P(A)>0P(B)>0 P(A;B)=P(A)P(B A)=P(B)P(A B) S AB P(B)=P(B A)P(A) +P(B AÇ )P(AÇ ) P(B)=P(B;S) P(A;B) P(A) P(B;A) P(B) P(B)=P(B;(A'AÇ )) P(B)=P(B;A)+P(B;AÇ ) P(B)=P(B A)P(A)+P(B AÇ )P(AÇ ) S A B 2 P(A;B) P(A;B) P(B A)= P(A B)= P(A) P(B) P(A;B)=P(A)P(B A)=P(B)P(A B) AB P(A;B)=P(A)P(B A)=P(B)P(A B) P(A)>0P(B)>0) S S A A A B A A A A A A B S=A 'A 'A P(A i )>0(i=123) P(B)=P(B A )P(A ) +P(B A )P(A )+P(B A )P(A ) P(B)=P(B;S)=P(B;(A 'A 'A )) P(B)=P(B;A )+P(B;A )+P(B;A ) P(B)=P(B A )P(A )+P(B A )P(A ) +P(B A )P(A ) n A A ya«a 'A 'y'a«=s P(A )>0 (i=12yn) P(B)=P(B A )P(A )+P(B A )P(A ) y+p(b A«)P(A«) 172
100 2 101 3 P(A;B)=P(B)P(A B) =0.5_0.3=0.15 3 AB P(B A)=0.2P(A)=0.6 P(A;B)=P(A)P(B A)=0.6_0.2=0.12 AB P(B)=0.5P(A B)=0.3 P(A;B) 101 4 102 1 A B P(A)=;1 0;P(B A)=;9@; P(A;B)=P(A)P(B A) P(A;B)=;1 0;_;9@;=;1 5; 2 5 5 2 2 A B P(A)=;1 0;=;2!;P(B A)=;9$; P(A;B)=P(A)P(B A) P(A;B)=;2!;_;9$;=;9@; yy` 4 10 3 1 1 A B P(A)=;2!;P(A;B)=;3!; P(B A)= P(A;B) P(A) ;3!; P(B A)= =;3@; ;2!; 2 A B P(A)=;1 5;P(B A)=;1 4; P(A;B)=P(A)P(B A) P(A;B)=;1 5;_;1 4;=;3 5; 2 15 4 2 1 AB P(A)=0.7 P(B)=0.8 P(A;B)=0.5P(B A) P(A B) P(B A)=;7%;P(A B)=;8%; 2 AB P(B A)=;3!; P(A B)=;5@; P(A'B) P(B) P(A;B)=P(A)P(B A)=P(B)P(A B) P(A;B)=;3!;P(A)=;5@;P(B) P(A)=;5^;P(B) P(A'B)=P(A)+P(B)-P(A;B) P(A'B)=;5^;P(B)+P(B)-;5@;P(B)=;5(;P(B) ;5(;P(B) P(A'B) = =;5(; P(B) P(B) 173
2 2 3 «C p q«(q=1-p) 1 2 A B 1 P(B A)=;5@;P(B AÇ )=;5@;P(B)=;5@; P(B A)=P(B AÇ )=P(B) A B P(B A)=P(B AÇ )=P(B) AB P(A;B)=P(A)P(B A)=P(A)P(B)(P(A)>0) 2 3 A B 1 2 _ 2 P(A;B) 5 5 P(B A)= = =;5@; P(A) 2 5 3 _ 2 P(AÇ;B) 5 5 P(B AÇ )= = =;5@; P(AÇ ) 3 5 2 2 _ 1 P(A;B) 5 4 P(B A)= = =;4!; P(A) 2 5 3 _ 2 P(AÇ ;B) 5 4 P(B AÇ )= = =;2!; P(AÇ ) 3 5 A B AB A B AB 174
3 AB AB B A P(B A)=P(B AÇ )=P(B) AB P(A;B)=P(A)P(B A) P(B A)=P(B) P(A;B)=P(A)P(B) P(A;B)=P(A)P(B) A B P(A;B) P(A)P(B) P(B A)= = =P(B) P(A) P(A) A B P(B A C P(AÇ ;B) P(B)-P(A;B) )= = P(AÇ ) 1-P(A) P(B)-P(A)P(B) = =P(B) 1-P(A) AB AB P(A;B)=P(A)P(B) P(A)>0P(B)>0 3 2 A B P(B A)=;4!;P(B AÇ )=;4@;=;2!; P(B A)+P(B AÇ ) AB P(B A)+P(B AÇ ) AB p.104 p.107 AB P(A;B)=P(A)P(B) (P(A)>0P(B)>0) P(A;B)+P(A)P(B) A B S={1 2 3 4} A={1 2} B={2 3} A;B={2} P(A)=;2!;P(B)=;2!;P(A;B)=;4!; P(A;B)=P(A)P(B) A B 1 A3 B A B A={1 3 5} B={3 6} P(A)=;2!;P(B)=;3!;P(A;B)=;6!; P(A;B)=;6!;=P(A)P(B) A B ABC A B B C C A P(A;B)=P(A)P(B) P(B;C)=P(B)P(C) P(A;C)=P(A)P(C) P(A;B;C)=P(A)P(B)P(C) A6 B C P(A)=;2!;P(B)=;3@;P(C)=;2!; P(A;B)=;3!;P(B;C)=;3!;P(A;C)=;6!; P(A;B;C)=P(A)P(B)P(C)=;6!; P(A;B)=P(A)P(B) P(B;C)=P(B)P(C) P(A;C)+P(A)P(C) A B B C A C A B P(A;B)=P(A)P(B) p.104 P(AÇ;BÇ )=P((A'B)Ç )=1-P(A'B) P(AÇ;BÇ )=1-{ P(A)+P(B)-P(A;B)} P(AÇ;BÇ )=1-{ P(A)+P(B)-P(A)P(B)} P(AÇ;BÇ )=(1-P(A))(1-P(B)) P(AÇ;BÇ )=P(AÇ )P(BÇ ) AÇ AB BÇ A Ç B AB Ç A B A Ç B Ç A BÇ AÇ B 4 3 175
1 1 10 10 A3 B5 C A BB CA C 1234 1 3 A P(A)=;6!;P(AÇ )=;6%; 5 3 2 C =10 1 2 3 345 3 ;6!;_;6!;_;6%;_;6%;_;6%;={;6!;} {;6%;} C C {;6!;} {;6%;} n 3 1 2 3 1 2 3 [ r «C {;6!;} {;6%;} «2 3 3 1 2 3 3 [ 4 A p n A r «C p q«(q=1-pr=012yn) 5 3 2 A ;3!; 6 A 2 C {;3!;} {;3@;} 103 1 105 1 A={246810}B={369}C={510} P(A)=;2!;P(B)=;1 0;P(C)=;5!; AB A;B={6} P(A;B)=;1 0; P(A;B)+P(A)P(B) A B BC B;C=u P(B;C)=0 P(B;C)+P(B)P(C) B C AC A;C={10} P(A;C)=;1 0; P(A;C)=P(A)P(C) A C 4 A p A 1-p n A r «C p_p_y_p_q_q_y_q=p q«r (n-r) (q=1-p) ( \ { \ 9 ( \ { \ 9 n A r «C p q«(q=1-pr=012yn) «C p q«p+q=1 (a+b)««c a b«a p n A r (a+b)«a b«176
104 3 105 2 ;2!; ;2!; C {;2!;} {;2!;} fl =;6@4);=;1 6; 2 6 3 2 3 30 % 3 4 3 1 3 ;1 0; 4 ;1 0; 105 4 106 1 ;3!; 1-[ Cº {;3!;} {;3@;} fi + C {;3!;} {;3@;} ] 3 3 3 4 C {;1 0;} {;1 0;} + C {;1 0;} {;1 0;} = 756 + 81 = 837 yy` 10000 10000 10000 1 1 1- Cº {;1 0;} {;1 0;} = 7599 yy` 10000 2 4 5 2 32 80 =1-{ + } 243 243 = 131 243 1 A5 B A B 2 5 6 2 6 4 1 A={235}B={56}A;B={5} P(A)=;2!;P(B)=;3!;P(A;B)=;6!; P(A;B)=P(A)P(B) A B 2 6 ;3@; 6 2 C {;3@;} {;3!;} 40 = 243 6 4 1- C {;3@;} fi {;3!;} 32 =1- = 211 243 243 1 12 A456 B C AB C A 2 1 3 3 3 4 1 3 ;3!; 3 ;3@; ;3!; C {;2!;} {;2!;} +;3@; C {;2!;} {;2!;} =;3!;_;8#;+;3@;_;1 6; =;8!;+;6!;=;2 4; 177
1 P(A;B)=P(A)P(B A) P(A'B)=;4#; =;3!;_;2!;=;6!; P(A'B)=P(A)+P(B)-P(A;B) ;4#;=;3!;+P(B)-;6!; - 2 1 AB P(A)=;3!;P(B A)=;2!;P(A'B)=;4#; P(B) 2 60 % ;3!; ;4!; P(B)=;4#;-;3!;+;6!; =;1 2; 3 3 5 2 20 % 40 % 60 % 10 % 30 % 40% 0.2 =;3@; 0.3 30 % 70 % 100 % 4 AB P(A'B)=;6%;P(A)=;3!; P(BÇ ) 5 80 % 5 3 4 5 3 ;8#;_;7%;+;8%;_;7#;=;5#6);=;2!8%; ;1 0;=;5$; ;1 0;=;5!; C {;5$;} {;5!;} fi =10_{;5$;} _{;5!;} 4 P(A;B)=P(A)P(B) P(A'B)=P(A)+P(B)-P(A) P(B) ;6%;=;3!;+P(B)-;3!;P(B) P(B)=;4#; P(BÇ )=1-P(B) 128 = 625 C {;5$;} {;5!;} fi + C {;5$;} fi {;5!;} fi fi =5_{;5$;} _{;5!;}+{;5$;} fi _{;5!;} = 2304 3125 =;4!; 178
1 4!_2 =;5@; (6-1)! n 5! 3! 6! 3!2! =;3!; n 1-;3!;=;3@; 1 6 banana 6 n 2 5 4 3 4 3 3 2 C C C + C C C + C C C C 120+90+60 = =;4@9&5); 495 =;1 1; 3 {1-;4!;} {1-;3@;} {1-;2!;}=;4#;_;3!;_;2!; =;8!; 1-;8!;=;8&; ;4!;;3@;;2!; 4 AB P(A)=0.4P(B)=0.5P(A'B)=0.6 P(A B) P(AÇ BÇ ) 4 P(A;B)=P(A)+P(B)-P(A'B) =0.4+0.5-0.6 =0.3 P(A;B) P(A B)= P(B) P(A B)= P(A B)=;5#; 0.3 0.5 P(AÇ BÇ )= P(AÇ BÇ )= P(AÇ BÇ )= P(AÇ BÇ )=;5$; P(AÇ ;BÇ ) P(BÇ ) 1-P(A'B) 1-P(B) 0.4 0.5 179
5 ;2!;;3!;;6!; ;2!; ;2!;+;3!; ;3!;+;6!; ;6!;=;4!;+;9!;+;3 6; =;1 8; 5 2 6 A B 6 A 3 5 B B 4 3 A A B ;8#;_;8%;=;6!4%; A B B B B ;8%;_;8$;=;6@4); ;6!4%;+;6@4);=;6#4%; 7 7 1 1 ;6!; ;6%; ;6%; ;6!;={;6%;} ;6!; ;6%; ;6%; ;6%; ;6%; ;6!;={;6%;} ;6!; n {;6%;} 2n-2 ;6!; ;6!; {;6%;} 2 ;6!;+{;6%;} ;6!;+{;6%;} ;6!; ;6!;+y= 1-{;6%;} =;1 1; A A A A 180
A 4 A A A B B B 5 A B B A A B 6 _ A B A B A 7 A _ A A ;2!; A {;2!;_;2!;}+{;2!;_;2!;_;2!;}+{;2!;_;2!;_;2!;_;2!;} +{;2!;_;2!;_;2!;}+{;2!;_;2!;_;2!;_;2!;} +{;2!;_;2!;_;2!;_;2!;}=;1!6!; AB 7 4 5:2 A 2 1 AB 1 k k 1 C {;2!;} 1 {;2!;} k-1 =k {;2!;} k (k=1 2 3 4 5 6) ;6!; ;K6+! ;6!;k {;2!;} k =;6!;[1 {;2!;} +2 {;2!;} +3 {;2!;} +4 {;2!;} +5 {;2!;} fi +6 {;2!;} fl ] ;K6+! ;6!;k {;2!;} k =;6!;{;2!;+;2!;+;8#;+;4!;+;3 2;+;3 2;} ;K6+! ;6!;k {;2!;} k =;6!;_;3^2);=;1 6; 7k A 5k_;1!6!;+2k_;1 6;=;1^6%;k ABC 1 1 1 B 7k-;1^6%;k=;1$6&;k AB=6547 1 1 2 2 3 1 2 2 3 3 2 2 3 3 3 A B C 2 1 A A ;3@; A {;3@;_;3@;}+{;3@;_;3!;_;3@;}+{;3@;_;3!;_;3!;_;3@;} +{;3!;_;3@;_;3@;}+{;3!;_;3@;_;3!;_;3@;} +{;3!;_;3!;_;3@;_;3@;}=;9*; 7k A 5k_;9*;+2k_;9!;=;; 3 ;;k B ABC 1 123 A ;6#;;6@;;6!; ;6#; 1 {;2!;} +;6@; 2{;2!;} +;6!; 3{;2!;} =;4@8#; B ;3!;;3!;;3!; ;3!; 1 {;2!;} +;3!; 2{;2!;} +;3!; 3{;2!;} =;2!4!; {=;4@8@;} C ;6!;;6@;;6#; 7k-;; 3 ;;k=;3&;k AB=21 ;6!; 1 {;2!;} +;6@; 2{;2!;} +;6#; 3{;2!;} =;1 6;{=;4@8!;} ABC 6 4 181
1 1 12 12 2 3 4 AB P(A)=;4!;P(B)=;5@; P(AÇ ;BÇ ) ;3!; ;5@; ;2!; ;3@; ;4#; 2 1 8 8 4 3 ;7!; ;6!; ;5!; 5 5 2 5 ;4!; ;3!; 3 A B P(A'B)=;5#;P(B A)=;2!; P(A B)=;4!;P(A;B) 6 50% 10 10 80 182
1 2 A3 B A;B 6 A={24681012}B={36912}A;B={612} P(A)=;2!;P(B)=;3!;P(A;B)=;6!; P(A'B)=P(A)+P(B)-P(A;B) =;2!;+;3!;-;6!;=;3@; 2 8! C = =70 4!4! 3 45678 5 3 5! C = =10 2!3! C C =;7!0);=;7!; 3 P(A;B) P(B A)= =;2!; P(A) P(A B)= P(A;B)=;2!; P(A) P(A;B) P(B) =;4!; P(A;B)=;4!; P(B) P(A)=;2!; P(B) P(A'B)=P(A)+P(B)-P(A;B) P(B)=;2!5@; =;2!; P(B)+P(B)-;4!; P(B) =;4%; P(B)=;5#; P(A;B)=;4!; P(B)=;4!;_;2!5@;=;2 5; yy yy 4 AB AÇBÇ P(AÇ;BÇ )=P(AÇ )P(BÇ ) P(AÇ;BÇ )=(1-P(A))(1-P(B)) P(AÇ;BÇ )={1-;4!;}{1-;5@;}=;4#;_;5#;=;2ª0; 5 3 2 yy! A B A B A O X O X X ;5@;_;4#;_;3!;_;2!;=;2 0; ;5@;_;4#;_;3!;_;2!;=;2 0; O X X X O ;5@;_;4#;_;3@;_;2!;=;1 0; ;5@;_;4#;_;3@;_;2!;=;1 0; X X O X O ;5#;_;4@;_;3@;_;2!;=;1 0; ;5#;_;4@;_;3@;_;2!;=;1 0; yy ;2 0;+;1 0;+;1 0;=;4!; yy 6 10 % 60 % 30 % ;2!;10 8 yy ºC {;2!;} 8 {;2!;} 2 + ºCª {;2!;} 9 {;2!;} 1 + ºC º {;2!;} 10 {;2!;} 0 45 10 1 56 7 = + + = = yy 1024 1024 1024 1024 128 7 121 1- = yy 128 128 10 8 20 % 60 % 20 % 183
1 4 6 2 2 ;1 5; ;1 5; ;5!; ;1 5; ;3!; 4 xy x+y=30 (xy) xy>125 q p+q pq p 32 36 40 44 48 2 63 4 5 S={123456}A={135} A B 3 4 5 5 2 6 A={23456} pqr pq+r 184
1 A B P(A;B)=P(A)P(B A) P(A;B)=;1 0;_;9#;=;1 5; 2 3 ;3!; C {;3!;} 4 {;3@;} 2 =;2 4º3; 3 0 1 A 2 A C Cº_ C C _ C P(A)= + ªC ªC 5 A B AB P(A)=;2!; P(A;B)=P(A)P(B) P(A;B)=;2!;P(B) A={135} P(A;B) ;6!;;3!;;2!; ` P(A;B)=;6!; P(B)=;3!;!` B C _ C =9 ` P(A;B)=;3!; P(B)=;3@;!` B C _ C =9 ` P(A;B)=;2!; P(B)=1(B=S)!` B 1 ` ` ` B 9+9+1=19 =;12!6;+;1 2º6;=;1 2 6;=;6!; yy P(A C )=1-P(A)=1-;6!;=;6%; 1 2 yy 70 % 30 % 6 pq+r pqr pqr yy ` pq ;5@;_;5@;=;2 5;!`pqr ;2 5;_;5#;=;1 2 5; 4 x+y=30 (xy) 29 y=30-x xy=x(30-x)>125 x -30x+125<0(x-5)(x-25)<0 5<x<25 xy>125 (xy) 19 ;2!9(; p=29q=19 p+q=48 `pq 1-;2 5;=;2@5!;!`pqr ;2@5!;_;5@;=;1 2 5; yy ;1 2 5;+;1 2 5;=;1 2 5; yy pq+r 30 % 60 % 10 % 185