(CIP) /,,. :,2013.9 ISBN978 7 5628 3622 3 Ⅰ.1 Ⅱ.1 2 3 Ⅲ.1 2 Ⅳ. 1021 44 CIP (2013) 178704 / / / / / : 130,200237 :(021)64250306( ) (021)64252174( ) :(021)64252707 :press.ecust.edu.cn / /787mm 1092mm 1/16 /16.75 /445 /2013 9 1 /2013 9 1 /ISBN978 7 5628 3622 3 /36.00 : press@ecust.edu.cn e.weibo.com/ecustpress htp://shop61951206.taobao.com
,,,,.,,. : 1.,.,,,,.,,,,. 2.,.,,.,,. 3.,.,,,.,,.,. 1 4, 5 6, 7 8.,.,,.,,.
1 1 1.1 1 1.2 1 1.3 2 1.4 7 1.5 ( ) 16 2 33 2.1 33 2.2 33 2.3 35 2.4 38 2.5 ( ) 49 3 74 3.1 74 3.2 74 3.3 77 3.4 82 3.5 ( ) 99 4 128 4.1 128 4.2 128 4.3 129 4.4 130 4.5 ( ) 134 5 143 5.1 143 5.2 143 5.3 144 1
5.4 147 5.5 ( ) 151 6 161 6.1 161 6.2 161 6.3 162 6.4 165 6.5 ( ) 173 7 191 7.1 191 7.1 191 7.3 192 7.4 195 7.5 ( ) 202 8 220 8.1 220 8.2 220 8.3 221 8.4 224 8.5 ( ) 232 260 2
1 1.1 1.,,. 2.,. 3.,. 4.,. 5., (Bayes),. 6.,. 1.2 Ω A B A=B AB A B A-B A B= A=Ω-A 1
1.3 1.3.1 1 1,, 2
1, :1, ;2 ;3, ω ( ), Ω ω, Ω, A, B, A,B, 2 A A, ω A, Ω, 1.3.2 1 2 A B, A B, B A, A B B A Ω A Ω A B,B A, A B, A=B A(B) Ω A B A B, A B A+B A B Ω A B A B, A B AB A B Ω 3
A B A B, A-B A B Ω A-B=A-AB=A(Ω-B) ( ) A B, AB=, A B A B Ω ( ) A, Ω-A A, 췍 A A 췍 A=Ω-A, 췍 A=A, A B :1AB= ;2A+B=Ω A Ω Ω E,B1,B2,,Bn E, 1 BiB j=,i j,i,j=1,2,,n; 2 B1 B2 Bn=Ω, B1,B2,,Bn 1 3 ( ) A B=B A,A B=B A (A B) C=A (B C),(A B) C=A (B C) A (B C)=(A B) (A C),A (B C)=(A B) (A C) A1 A2 An= A1 췍 A2 췍 An,A1 A2 An= 췍 A1 췍 A2 췍췍 An 1.3.3 1 4 n, A m, fn(a)= m n A n 4
1 n, n A m, n, fn(a) p, p A, P(A)=p., Ω, (, ) SΩ(SΩ < ), ( ). A Ω,SA A, A P(A)= SA SΩ P( )=0 Ω={ω1 ὠ2, ὠn}, n ( ), ω1, ὠn ( ), P(ω1)=P(ω2)= =P(ωn)= 1 n, A P(A)= A Ω = A Ω = k n Ω,A, P(A) Ω A, : 1( ) P(A) 0; 2( ) P(Ω)=1; 3 ( ) A1,A2,,An,, 1 5 P( n=1an)= P(An) n=1 0 P(A) 1 A1,A2,,An, n n P( i=1ai)= P(Ai) i=1 A1,A2,,An, n n P( i=1ai) P(Ai) i=1 P( 췍 A )=1-P(A) P(A-B)=P(A)-P(AB) B A, P(A-B)=P(A)-P(B) B A, P(B) P(A) P(A B)=P(A)+P(B)-P(AB) P(A B C)=P(A)+P(B)+P(C)-P(AB) -P(AC)-P(BC)+P(ABC) P(A B) P(A)+P(B) 1.3.4 1 6 A,B (Ω,F,P),P(B)>0, P(A B)= P (AB) P(B) B, A 5
(1) : 1( ) P(A B) 0; 2( ) P(Ω B)=1; 3 ( ) A1,A2,,An,, P( (2) ( 1 5) B n=1an ) = P(An B) n=1 1 7 n P(A)>0, P(AB)=P(A)P(B A) P(A1A2 An-1)>0, P ( n i i=1a ) =P(A1)P(A2 A1)P(A3 A1A2) P(An A1A2 An-1) B1,B2,,Bn, P(Bi)>0(i=1,2,,n), A, n P(A)= P(Bi)P(A Bi) i=1 B1,B2,,Bn, P(Bi)>0(i=1,2,,n),P(A)>0, P(Bi A)= P (Bi)P(A Bi) n P(B j )P(A B j ) 1.3.5 j=1 1 8 n A1,A2,,An, : P(AiA j )=P(Ai)P(A j ), n A1,A2,,An P(AB)=P(A)P(B), A A B, A 췍 B, 췍 A B B, 췍 A 췍 B n A1,A2,,An, k(2 k n) : P(Ai 1 Ai 2 Ai k )=P(Ai 1 )P(Ai 2 ) P(Ai k ), n A1,A2,,An, n ( ) n, n n A1,A2,,An, m(1 m n), n 6
1 1 9 n A1,A2,,An ( ) P(A1 A2 An)=P(A1)P(A2) P(An) P (A1 A2 An)=1-P( 췍 A1 )P( 췍 A2 ) P( 췍 An ) =1-(1-P(A1))(1-P(A2)) (1-P(An)) 1.4,,,..,. 1 1 A,B,C, A,B,C : (1)A,B,C ; (2)A,B,C ; (3)A,B,C ; (4)A,B,C ; (5)A,B,C. :,. :(1)A,B,C, A,B,C ABC. (2) A,B,C, 췍 A, 췍 B, 췍 C, A,B,C 췍 A 췍 B 췍 C. (3)A,B,C A,B,C, ABC. (4) ( ),A,B,C A B C. (5) A,B,C A,B,C. A,B,C, A,B,C A 췍 B 췍 C 췍 AB 췍 C 췍 A 췍 BC, A 췍 B 췍 C 췍 AB 췍 C 췍 A 췍 BC; A,B,C 췍 A 췍 B 췍 C, A,B,C A 췍 B 췍 C 췍 AB 췍 C 췍 A 췍 BC 췍 A 췍 B 췍 C= 췍 A 췍 B 췍 A 췍 C 췍 B 췍 C=AB AC BC. 1 2 (2004 ) 4,,, 2 t0,, E, T(1) T(2) T(3) T(4) 4, E ( ). (A) { T(1) t } { } { } { } 0 (B) T(2) t 0 (C) T(3) t 0 (D) T(4) t 0 :. T(i) t { 0} 5-i t0, 2 t0,, E { T(i) t 0 },i=1,2,3, E { T(3) t } :(C). 0, E= T(3) t 1 3 A,B, ( ). (A)(A-B)+B=A (C)(A+B)-B=A { 0 }, (C). (B)AB+(A-B)=A (D)(AB+A)-B=A : ( ). 7
,,.,. A,. 1 3,, (A) A+B, (C) A-B, (D) A-B, (B) AB+A B=A, (B). 췍 A B A B A B A B (A) (B) (C) (D) 1 3 :(B). Ω ( SΩ), A ( ) SA, A P(A)= SA. SΩ 1 4,, 1 2,. : x,y. xy. : x,y ( : ), Ω={(x,y) 0 x 24,0 y 24}. 1 2, A, A={(x,y) 0 x-y 1,0 y-x 2}, 1 4, 24 2-1 P(A)= SA 2 ( 22 2 +23 2 ) = 0.121. SΩ 24 2 y y=x2 O y=x1 1 4 x 1 5 a,. : x y z, x+y+z=a,x>0,y>0,z>0. x>0, y>0,x+y<a( z=a-x-y>0)., ( 1 5 ) Ω={(x,y) x>0,y>0,x+y<a}. 8
1 y a a 2 A x+y=a Ω O a 2 a x 1 5, x+z>y,y+z>x,x+z>y, z=a-xy, x+y> a 2, x< a 2, y< a 2. 1 5 A. : 1 5 Ὠ a, SΩ= a 2 2 ; A,, SA = a2 8. /8, P(A)= SA = a2 SΩ a 2 /2 =1 4., A P(A)= A Ω =k n., : n r (1 r n)? r : n r C r n+r-1 n(n-1) (n-r+1) C r n.,. 1 6 1,2,,n, (1) ;(2).. :,, ;, n r., i i+1, i n-1,, 2(n-1). :(1), n(n-1), 2(n-1), 9
p1= 2 (n-1) n(n-1) =2 n. (2), n 2, (1), 1 7 0 1 2 3. p2= 2 (n-1) n 2. 4 4, 0 1 2 3, :, 4,.,. : Ai i,i=0,1,2,3. 4,, 4 4. A0,, A0 4!, P(A0)= 4! 4 4 = 3 32. A1 1, 1 2,2 1, C 1 4C 2 3, C 2 4 2!, A1 C 1 4C 2 3 C 2 4 2!, 1 P(A1)= C1 4C 2 3 C 2 4 2! 4 4 = 9 16. A2, 2 1 3 1 1, 2 2, 1 3 1 1 C 1 4C 1 3C 3 4, 2 2 C 2 4C 2 4, A2 C 1 4C 1 3C 3 4+C 2 4C 2 4, 2 P(A2)= C1 4C 1 3C 3 4+C 2 4C 2 4 4 4 = 21 64. A3 3,,, A3 C 1 4, 3 P(A1)= C1 4 4 4 = 1 64..,., P(A)=1-P( 췍 A ); P(A+B)=P(A)+P(B)-P(AB); P(A-B)=P(A)-P(AB). 1 8 (2009 ) A B, ( ). (A)P( 췍 A 췍 B )=0 (B)P(AB)=P(A)P(B) (C)P(A)=1-P(B) (D)P( 췍 A 췍 B )=1 :. 1 3 1 5 P( 췍 A 췍 B )=1-P(A B), P(A B) 0, (A);(B) A B, A B, P(AB)=0, P(A) P(B) 0, (B); A B, (C), ; 10
1 P( 췍 A 췍 B )=1-P(AB)=1, (D). :(D). 1 9 A B C, P(A)=a,P(B)=2a,P(C)=3a,P(AB)=P(AC)= P(BC)=b, :a 1 4, b 1 4. :. P(A B)= P(A)+P(B)-P(AB)=3a-b,P(A C)=4a-b,P(B C)=5a-b, ABC AB, P(ABC) P(AB)=b, P(A B C)=6a-3b+P(ABC) 6a-2b, 1, 5a-b 1. AB A, b=p(ab) P(A)=a, b=p(ab) P(B)=2a,b=P(AC) P(C)=3a, b a, 4a 5a-b 1,. : P(B C)=P(B)+P(C)-P(BC)=5a-b 1 b=p(ab) P(A)=a, 4a 5a-b 1, b a 1 4. 1 10 n-1 1,,, k? :.,, k, k-1,,. : A k, k P( 췍 A )= ( n-1) k-1 1 n k. P(A)=1- ( n-1) k-1 n k. P(A B), P(B) 0. P(A B)= P(AB) P(B), B A, B P(A B),. 1 11 a,b, ( ). (1), ; (2), ; (3),. : (1), ; (2) P(A1 A2)= P (A1A2) P(A2), P(A1A2) P(A2) ; (3), A1 A2.,,, P(A1A2 A1 A2)= P (A1A2) P(A1 A2), P(A1 A2). : Ai i,i=1,2, (1)P(A2 A1)= a-1 a+b-1 ; 11
(2)P(A1 A2)= P (A1A2) P(A2) = a (a-1) a (a+b)(a+b-1) a+b = a-1 a+b-1 ; (3)P(A1A2 A1 A2)= P (A1A2) P(A1 A2) = P(A1A2) P(A1)+P(A2)-P(A1A2) = a (a-1) (a+b)(a+b-1) = a-1 a+2b-1. æ a a+b + a a+b - a (a-1) ö ç è (a+b)(a+b-1 ) ø 1 12 (2012 ) A,B,C,A,C,P(AB)= 1 2, P(C)= 1 3, P(AB 췍 C)=. :, P(AB 췍 C). A,C, A 췍 C, P(AB 췍 C)=P(AB)= 1 2, : 3 4. P(AB 췍 C)= P (AB 췍 C) P( 췍 C ) =3 4. P(AB)=P(A B)P(B)=P(B A)P(A)( ), A,B, P(AB)=P(A)P(B)., P(ABC)=P(A BC)P(BC)=P(A BC)P(B C)P(C).. 1 13, p(0<p<1), :,,,,.,,,,. : C, C. Bk k,k=0,1,2,3, C=B1 B2 B3,BiB j =,i j,i,j=1,2,3. 3 P(C)= P(Bi). i=1 B1, A j j,j=1,2,3, B1=A1 췍 A2 췍 A3+ 췍 A1A2 췍 A3+ 췍 A1 췍 A2A3. B2=A1A2 췍 A3+A1 췍 A2A3+A1A2 췍 A3,B3=A1A2A3.,P(A j )=p,j=1,2,3 A1,A2,A3, P(C)=3pq 2 +3p 2 q+p 3 = 3pq+p 3, q=1-p.. :,, C=A1 A2 A3, 12
1 3 P(C)= P(Ai)-P(A1A2)-P(A1A3)-P(A2A3)+P(A1A2A3) i=1 =3p-3p 2 +p 3 =3pq+p 3. : A1,A2,A3, A1,A2,A3, C, C=A1 A2 A3=A1 췍 A1A2 췍 A1 췍 A2A3, P(C)=P(A1)+P( 췍 A1A2)+P( 췍 A1 췍 A2A3) =P(A1)+P( 췍 A1 )P(A2)+P( 췍 A1 )P( 췍 A2 )P(A3) =p+pq+pq 2 =3pq+p 3. :, 췍 C=A1 A2 A3= 췍 A1 췍 A2 췍 A3, P(C)=1-P( 췍 C )=1-P( 췍 A1 )P( 췍 A2 )P( 췍 A3 ) =1-q 3 =(p+q) 3 -q 3 =3pq 2 +3p 2 q+p 3 =3pq+p 3 ( ).,, k. 1 14 (2003 ), :A1={ }, A2={ },A3={ },A5={ }, (A)A1,A2,A3 ; (B)A2,A3,A4 ; (C)A1,A2,A3 ; (D)A2,A3,A4. :.,., P(A1)= 1 2, P(A2)= 1 2, P(A3)= 1 2, P(A4)= 1 4, P(AiA j )= 1 4, i j,i,j=1,2,3,4 P(A1A2A3)=0. P(A1A2A3)=0, (A); P(A2A4)= 1 4 P (A2)P(A4) (B) (D), (C). :(C). 1 15 A,B,C, ABC=, (1) P(A)=P(B)=P(C)=x, x ; (2) P(A)=P(B)=P(C)< 1 2, P(A B C)= 9 16, P(A). :(1), x.(2). :(1) A,B,C, P(AB)=P(A)P(B)=x 2 ; P(BC)=P(B)P(C)=x 2 ; P(AC)=P(A)P(C)=x 2. ABC= P(ABC)=0. P(A-B-C)=P(A)-P(AB)-P(AC)+P(ABC)= x-2x 2 0, x 0 x 0.5, x 0.5. (2), 9 16 =P (A B C)=P(A)+P(B)+P(C)-P(AB)-P(BC)-P(AC)+P(ABC) 9 16 =3P (A)-3[P(A)] 2, P(A)=0.25 P(A)=0.75. 0 P(A) 0.5, 13
P(A)=0.25.. A, n P(A)= P(Bi)P(A Bi)., i=1, A Bi P(Bi A)= P(Bi)P(A Bi) n j=1 P(B j )P(A B j ) (i=1,2,,n). Ω B1, B2,,Bn, Bi P(Bi), ( ) P(A Bi)(i=1,2,,n). 1 16, 0.03, 0.02.,. (1). (2),. :, ( ),, (1); (2),,. : Ai i,(i=1,2), B. (1) P(A1)= 2 3, P(A2)= 1 3, P(B A1)=1-P( 췍 B A1 )=0.97,P(B A2)=1-P( 췍 B A2 ) =0.98, P(B)=P(A1)P(B A1)+P(A2)P(B A2)= 2 3 0.97+1 3 0.98=0.973. (2),, P(A2 췍 P(A2 B)= P 췍 (A2 B) P( 췍 B ) =P (A2)P( 췍 B A2 ) P( 췍 B ). 췍 B). P(A2)= 1 3, P( 췍 B A2 )=0.02, (1) P(B)=0.973,,P( 췍 B )=1-P(B)= 1-0.973=0.027,, 1 17 췍 P(A2 B)= P 췍 (A2 B) P( 췍 B ) =P (A2)P( 췍 B A2 ) P( 췍 B ) 0.02 1 3 = 0.027 =20 81 0.247. 100., 10,,. 4, i(i=0,1,2,3,4) :. 0 1 2 3 4 0.1 0.2 0.4 0.2 0.1 (1) ;(2) i(i=0,1,2,3,4) 14
1. 3,4. :, : A, Bi (100 ) i,i=0,1,2, P(B0)=0.1,P(B1)=0.2,P(B2)=0.4,P(B3)=0.2,P(B4)=0.1. 100-i C 10 100 P(A Bi)= C10 (1),i=0,1,2,3,4. 4 P(A)= P(Bi)P(A Bi)=0.8142. i=0 (2) i(i=0,1,2,3,4) P(B0 A)= P (B0)P(A B0) 4 =0.123, P(B j )P(A B j ) j=0 P(B1 A)=0.221,P(B2 A)=0.397, P(B3 A)=0.179,P(B4 A)=0.080. : P(Bi A).,, (100 ) i, 0 1 P(Bi), 1 P(Bi),. 1 18 ( ),, p. ); (1) ( (2) n ; (3), ; (4), 0.1,, (3) ; (5), ( p=0.5). :, p,. : Ai i (i=1,2,,n),p(ai)=p. (1), A1+ 췍 A1A2, P(A1+ 췍 A1A2)=p+(1-p)p=2p-p 2. (2) n,,, n, A1+ 췍 A1A2+ + 췍 A1 췍 An-1An, P(A1+ 췍 A1A2+ + 췍 A1 췍 An-1An)=p+(1-p)p+ +(1-p) n-1 p=1-(1-p) n. 15
(3),,, P( 췍 A1 췍 A2 췍 A3 )=(1-p) 3. (4) (3),,,. B, B 췍 A1 췍 A2 췍 A3+ 췍 B, P(B 췍 A1 췍 A2 췍 A3+ 췍 B)=0.1(1-p) 3 +0.9. (5), P( )= ( 1-p) 3 0.1 (1-p) 3 0.1+0.9, p=0.5, P( )= ( 1-0.5) 3 0.1 (1-0.5) 3 0.1+0.9 =0.0137. 1.5 ( ) 1 1 : (1),. (2) 5, 1 2 3 4 5, 3, 1. (3) 1,. :(1)Ω={(, ),(, ),(, ),(, )},A={(, ),(, )}. (2)Ω={(1,2,3),(1,2,4),(1,2,5),(1,3,4),(1,3,5),(1,4,5),(2,3,4),(2,3,5),(2, 4,5),(3,4,5)}, A={(1,2,3),(1,2,4),(1,2,5),(1,3,4),(1,3,5),(1,4,5)}. (3)Ω={(x,y,z) x+y+z=1,x,y,z>0}. 1 2, A, B 3 : (1) A B. (2) 췍 A, 췍 B,A B,AB,A B,. : :1,2,3,4,5,6, (1)A={2,4,6},B={3,6}; (2) A= 췍 {1,3,5} ; B={1,2,4,5} 3 ; 췍 A B={2,3,4,6} 2 3 ; AB={6} 2 3 ; A B={1,5} 2, 3. 1 3, A 40, B, C 40. x,y A B C, : (1)ABC;(2)A-AB;(3)A BC. 췍 : A,B,C, ( 1 3 ): 16
1 1 3 A, B, C. : (1)ABC 40 ; (2)A-AB 40 ; (3)A 췍 BC 40. 1 4, :A 췍 C B. : A 췍 C 40, 40,, A 췍 C B. 1 5 1 2 3 4 24,. Ai i i,i=1,2,3,4, :A1A2A3 A4 A1A2 췍 A3 췍 A4. : A1={1234,1243,1324,1342,1423,1432}; A2={1234,1243,3214,3241,4213,4231}; A3={1234,2134,1432,4132,2431,4231}; A4={1234,1324,2134,2314,3124,3214}; A1A2A3={1234} A4,A1A2 췍 A3={1243} 췍 A4. 1 6,, A=, B=, C=, D=, ξ, ξ A B C D. A B C D?? :A={ ξ =1},B={ ξ 1},C={ ξ =2},D={ ξ =0}. :A C,A D,B D,C D, :B D. 1 7 A B, : 췍 A 췍 B. : A B, AB=,A B=Ω, 췍 A 췍 B=Ω, 췍 A 췍 B=, 췍 A 췍 B. 1 8 1 5, : P(A1+A2)=P(A1)+P(A2)-P(A1A2). : 1 5 A1,A2 P(A1)= 6 24, P(A2)= 6 24, P(A1+A2)= 10 24, P(A1A2)= 2 24, 17
P(A1+A2)= 10 24 =6 24 +6 24-2 24 =P (A1)+P(A2)-P(A1A2). 1 9 0<y< 4x-x 2,, x π 6. :, Ω= { ( x, y ) (x-2) 2 +y 2 4,y 0 }, SΩ =2π. A x π 6, SA = 3 æ 0 4x-x2-3 ö ç x dx= 4π è 3 ø 3-3. y A O x 1 9 x π 6 P(A)= SA = 2 3-3 2π. SΩ 1 10 n,, ( ). 1 2, n, n 1 n. : Ai i. i, P i n, i (i-1) (n-1),, Ai P i-1 n-1, i P(Ai)= Pi-1 n-1 = ( n-1)(n-2) (n-i+1) n(n-1) (n-i+2)(n-i+1) =1 n. P i n 1 11 10,. : A 10.10 10!, 3 :13 3!;27 7!; 3 7,3 8,, 8 3! 7!, P(A)= 8 3! 7! = 1 10! 15. 1 12 17, 10, 4, 3,,, 4 3 2,? :. A 4 3 2. 18
1 C 9 17, A C 4 10C 3 4C 2 3, 4 3 2 P(A)= C4 10C 3 4C 2 3 = 252 C 9 17 2431. 1 13,,,, : (1) ; (2) ; (3), ; (4). : (1)~(3). A, B, C,. C 2 10. (1) C 2 8, P(A)= C2 8 = 28 C 2 10 45. (2) C 2 2, P(B)= C2 2 = 1 C 2 10 45. (3),,, P(C)= C1 8C 1 2 = 16 C 2 10 45. (4) D, P(D)= 2 10 =1 5. 1 14 A,B A B p q r, :P(AB),P(A 췍 B),P( 췍 AB ),P( 췍 A 췍 B ). : P(A B)=P(A)+P(B)-P(AB), P(AB)=P(A)+P(B)-P(A B)=p+q-r, P(A 췍 B)=P(A B)-P(B)=r-q, P( 췍 AB )=P(A B)-P(A)=r-p, P( 췍 A 췍 B )=1-P(A B)=1-r. 1 15 P(A)=0.7,P(A-B)=0.3, P(AB). : P(A-B)=P(A)-P(AB), P(AB)=P(A)-P(A-B)=0.7-0.3=0.4, P(AB)=1-P(AB)=1-0.4=0.6. 1 16 P( 췍 A )=0.3,P(B)=0.4,P(A 췍 B)=0.5, P(B A 췍 B). :,P( 췍 AB )=P(A-B)=P(A)-P(AB), P(A)=1-P( 췍 A )=1-0.3=0.7, P(AB)=P(A)-P(A 췍 B)=0.7-0.5=0.2, P( 췍 AB )=P(B)-P(AB)=0.4-0.2=0.2. 19
P(B A 췍 B)= P [B(A 췍 B)] P(A 췍 B) = P (AB) 1-P( 췍 AB ) = 0.2 1-0.2 =0.25. 1 17,, ;,? : Ai i, B, B=A1 + 췍 A1A2 + 췍 A1 췍 A2A3,,, P (A1 )= 1 10, P (A2 췍 A1 )= 1 9, P(A3 췍췍 A1A2)= 1 8, P(B)=P(A1)+P( A1A2)+P( 췍 A1 췍 A2A3) 췍 =P(A1)+P(A2 A1)P( 췍 A1 췍 )+P(A3 췍췍 A1A2)P( A2 췍 A1)P( 췍 A1 췍 ) = 1 10 +1 9 9 10 +1 8 8 9 9 10 =0.3.,, P(A1)= 1 5, P(A2 췍 A1)= 1 4, P(A3 췍 A1 췍 A2)= 1 3, P(B)=P(A1)+P( 췍 A1A2)+P( 췍 A1 췍 A2A3) =P(A1)+P(A2 췍 A1)P( 췍 A1 )+P(A3 췍 A1 췍 A2)P( 췍 A2 췍 A1)P( 췍 A1 ) = 1 5 +1 4 4 5 +1 3 3 4 4 5 =0.6. 1 18, A. A 65 81, A. : A p, Ai i A, A 췍 A1 췍 A2 췍 A3 췍 A4, p= 1 3. (1-p) 4 =P( 췍 A1 췍 A2 췍 A3 췍 A4 )=1-65 81. 1 19, 0.03, 0.02,,, ;,. 1 16. 1 20 A,B,C, 1 6 1 4 1 3,, : (1)? (2), B? : D. P(A)=P(B)=P(C)= 1 3, P(D A)= 1 6, P(D B)= 1 4, P(D C)= 1 3. 20
1 (1) P(D)=P(D A)P(A)+P(D B)P(B)+P(D C)P(C) = 1 6 1 3 +1 4 1 3 +1 3 1 3 =1 4. (2) 1 P(B D)= P (BD) P(D) =P (D B)P(B) P(D) = 3 1 4 = 1 1 3. 4 1 21 A,B,C, A B C : (1)A,B,C ; (2)A,B,C ; (3)A,B,C ; (4)A,B,C ; (5)A,B,C. 1 1. 1 22 A B, A B=A (B-AB), A,B,C. : A B C=(A B) C=(A B) (C-C(A B)), A B=A (B-AB) A B C=A (B-AB) (C-C(A B)). 1 23 R,, 3R. : 1 23, R,, A. A 3R PAQ, 4π 4π 3, 3 3R P= 2π =2 3. 1 24 a,. 1 5. 3R P A 3R 1 23 1 25 (p,q) Ω:[-1,1] [-1,1], x 2 +px+q=0 : (1) ; (2). :. Ω:[-1,1] [-1,1] SΩ=4. (1) x 2 +px+q=0, p 2-4q 0, (p,q) Ω x 2 +px+q=0 1 25(a). Q 21
[1],,.. :,2011. [2],.. :,2011. [3],,.. :,2004. [4]. (4 ). :,2003. [5].. :,2005. [6].. :,2002. [7].. :,2012. [8]R.V.Hogg&E.A.Tanis,ProbabilityandStatisticalInference.2001,PrenticeHal. [9],.. :,2002. [10].. :,2003. [11].. :,2003. [12],,.. :,2003. 260