. 80% E E E 0 0 E E 4
E E ω E E Ω E E Ω ={} E 0 0 =,, L, 0 E Ω= {,, L, 0} ω = ω = Ω= { ω, ω } E k k =,, L,, L E Ω= {,, L,, L} 4 E 4 t 0 t <+ E 4 Ω= { t 0 t <+ } 5 E 5 E 5 ()()()()( ). Ω ={()()()()}. ABC ω {ω }
Ω Ω A A Ω A B B A A B AB BA Ω= {,,,4,5,6} A ={,4} B ={,4,6} A B A B A A Ω Ω A B A B A B B A - B A Ω. A B AB BA A B A=B
4 AB A=B A B A B AB AB A B A B AB A B. A B Ω. A B A ={,} B ={, } C = A B={,, } { 4} A, A, L, A A, A, L, A A A L A U A = A, A, L, A, L A, A, L, A, L A A LA L U A = A B A B AB AB AB A B A B AB A B - A={,,5} B ={,} C = AI B = {} { }
A B Ω. AB A A L A I A A, A, L, A = I A A, A, L, A, L = 4 A B A B A B A B A B.4 B A Ω.4 A B A = {,, 4} B = {, } A B= {, 4} 5 A B AB AB = A B A B A + B 5
6 A B A B A B.5 B A Ω.5 AB = A, A, L, A A I A = j, j =,, L,. j A, A, L, A A, A, L, A A + A + L + A A B A A( B A) = 6 A A=Ω A A A A A Ω A A A A.6 A = A. A A Ω.6 A A A A A A A
AA = A+ A=Ω AB A B = AB Ω AB AB AB AB A = {, } B = {,5} A B B A A A = {, 4,5, 6} 7 A A A, A, L, A AA LA = Ω A A = j, j =,, L, j A, A, L, A Ω.7.7 Ω A = {, } B = {,5} C = {4,6} AB = BA AB = BA... AUBU C = ( AUB) UC = AU( BUC).. 7
8 ABC = ( AB) C = A( BC).... ( AB) C = ( AC)( BC)..4 ( AUB) C = ( AC) U ( BC)...5 4. AB = A B AB = AB..6 A, A, L, A U A = I A I A = U A = = = =...7 6 ABC ABC. B C A ABC ABC 4ABC 5ABC 6ABC. BC A ABC ABC ABC ABC ABCABCABC 4ABC ABCABCABC 5ABC ABCABCABCABC 6AB C ABC AB C.... 00 95 5.. 00 50.... A
A A A A f ( A ).. -. - A A 048 06 0.58 4040 048 0.5069 K 000 609 0.506 K 4000 0 0.5005 0.5 0.5. 0 f ( A). Ω f ( Ω ) =. AB f ( ) A B = f ( A ) + f ( B ). A A A A A. A. A.. A A p p A P( A ). P( A) = p. P( A) A.. 0.5. 9
0. A 0 PA ( )... P( Ω ) = P( ) = 0... A, A, L, A P ( AA LA ) = P( A) + P( A) + L + P( A )... A B P( A B ) = PA ( ) + P( B)...4 4 A B P( B A) = P( B AB) = PB ( ) PAB ( )...5 A B P( B A) = P( B) P( A)...6 5 A A P( A) = PA ( )...7 A U A = Ω AA = = P( Ω ) = P( A+ A) = PA ( ) + PA ( ). P( A) = P( A). PA ( A LA ) = P( AA L A ) = P( AA L A )...8 6 A B P( A B ) = P( A) + P( B) P( AB)...9.8 A, A, A PA ( AA ) = P( A) + P( A) + P( A) PAA ( ) PAA ( ) P( AA) + PAAA ( )...0 AA A A A AA A A AA A Ω.8
P( AA L A ) ( ) ( ) ( ) = P A P AA + P AA A j j k = < j < j< k + + ( ) PAA ( A) PAB = PB ( ) = 0.6 PA ( ) = 0.4 ( ) 0. L L. P( AB) P( AU B) P( AB ). AB + AB = B AB AB PAB ( ) + PAB ( ) = PB ( ) P( AB ) = P( B) P( AB) = 0.6 0. = 0. P( A) = P( A) = 0.4 = 0.6 P( AU B) = P( A) + P( B) P( AB) = 0.6 + 0.6 0. = 0.9 P( AB ) = P( AB) = P( AB) = 0.9= 0.. A B. A 45% B 5% 0%.. A = { A } B = { B } A B = { }. P( A B ) = PA ( ) + PB ( ) PAB ( ) = 0.45 + 0.5 0. = 0.7 AB + AB ={} P( AB + AB) = P( AB) P( AB) = 0.7 0. = 0.6... ω ω Ω { ω } { ω } P( ω ) = P( ω ) =... E
.. A m Ω A A m P( A ) = =... Ω A... A m P( A ).. 45 5. A ={ }. 50 C 50 C50 A C 45C5. CC 5 45 PA ( ) = 0.5 C50 N M N mm m m CMCN M p =. CN 8 4.. 4. A = {} A = {} A = { } A 4 = {} B = { }. C8 PA ( ) = = 0.55 C CC 8 4 PA ( ) = = 0.509 C PA ( ) = PA ( ) = PA ( ) = 0.55 = 0.745
4 B = A A. A 4 A 4 C PA ( = = 0.08 4 ) 4 C PB ( ) = PA ( ) + PA ( ) = 0.55 + 0.08 = 0.7. 4 a b A k 0 < k a+ b. a + b a + b ( a+ b)! A a k a+ b a+ b C( a+ b a )! C( )! ( ) a a + b PA= = a ( a+ b)! a+ b. k k a + b a + b A a k A a a PA ( ) = a + b. k k.. 4 ~00 5. A ={ }B ={ }C ={ 5 }. 50 PA= ( ) = 0.5 00 5 PA ( B C) = P( A) + P( B) + P( C) P( AB) P( AC) PBC ( ) + PABC ( ) 50 + + 0 6 0 6 + = = 0.74. 00.. 5 [,6] x x.
4 6 M M 5.. x [,6] x [, ] [, ] [,6] 5 M 5 π 5 π 5. Ω Ω. AΩ A A.. Ω S Ω Ω. AΩ Ω S A A S A S Ω PA ( ) =.. A. 7.. 0. Ω =[0,60]. A 0 A = [50,60]. SA 0 PA ( ) = = =. 60 6 S Ω.4. A P( A) B A P( A B )..
40 5 5 40.7 0..7?.7? A = {.7 } B = {}. P( A) B A P( A B ). 0 PA= ( ) = 0.55 PA ( B ) = = 0.8. 40 5 5 0 PB ( ) = PAB ( ) = 40 40 0 0 / 40 P( AB) PA ( B) = = =. 5 5/ 40 P( B). A B PB ( ) > 0 P( AB) PA ( B) =..4. PB ( ) B A A B. PA> ( ) 0 B A P( AB) PB ( A) = P( A) A B PAB ( ) = PBPA ( ) ( B), PB ( ) > 0.4. PAB ( ) = PAPB ( ) ( A), PA ( ) > 0. I P( A) > 0 = P( AAA ) = P( A) P( A A) P( A AA ). 4 PAAA ( LA) = PA ( ) PA ( A) PA ( AA ) LPA ( AA L A ). 44 8. A = { } PA= ( ) =. 6 5
6 B = { 8}. 6 = 6 AB = {(,6),(6,), (6, 4),(6,5),(6,6)}. 5 5 PAB ( ) 5 PAB ( ) = PBA ( ) = = 6 =. 6 PA ( ) 0% 8% 0.6. %. A = {} B ={}. PAB ( ) = PAPB ( ) ( A) = 0. 0.6 = 0. P( A B ) = P( A) + P( B) P( AB) = 0. + 0.8 0. = 0.6 PAB ( ) 0. PA ( B) = = = 0.67. PB ( ) 0.8. Ω A, A, L, A Ω AA = j A + A + L + A =Ω. j PA ( ) > 0 =,, L, B BA BA P( B) = P( A) P( B A)..4.5 = B = B Ω = B ( A + A + L+ A ) = BA + BA + L + BA.,, L, BA P( B) = P( BA + BA + L+ BA ) = P( BA ) = P( A ) P( B A ) = =. 70% 0%.
60% 0%.. A = {} B ={} B = {}. PB ( ) = 0.7 PB ( ) = 0. PAB ( ) = 0.6 PAB ( ) = 0.. A= AB+ AB P( A) = P( B) P( A B) + P( B) P( A B) = 0.7 0.6 + 0. 0. = 0.48. 4 4 4... B = {}. A =,, PA ( ) = PA ( ) = PA ( ) = 4 P( BA ) = PBA ( ) = PBA ( ) =. 6 6 6 P( B) = P( A) P( B A) + P( A) P( B A) + P( A) P( B A) 4 = + + = 0.5. 6 6 6 5 0. 0 0.80. 0.. 4.. B = {} A = { } = 0,, PA ( 0 ) = 0.8 PA ( ) = 0. PA ( ) = 0.. 4 4 C9 C8 PBA ( 0) = PBA ( ) = PBA ( 4 ) =. 4 C0 C0 P( B) = P( A0) P( B A0) + P( A) P( B A) + P( A) P( B A) 4 4 C9 C8 = 0.8 + 0. + 0. 0.94. 4 4 C0 C0 A, A, L, A Ω A P( A ) B A P( BA ) =,, L, PB ( ). B A =,, L, A B P( A B) 7
8. P( A B). Bayes A, A, L, A Ω PB ( ) > 0 PA ( ) > 0 =,, L, P( AB) P( A ) P( B A) PAB ( ) = = P( B) PB ( ) = PAPB ( ) ( A) =,, L,.4.6 P( A) P( B A) =. 6 5 P( AB 0 ) PA ( 0) PBA ( 0) 0.8 PA ( 0 B) = = = 0.85. P( B) PB ( ) 0.94 7 % % 60% 40%? B = {} A = {} A = {}. PA= ( ) 0.6 PA ( ) = 0.4 PBA ( ) = 0.0 PBA ( ) = 0.0. PB ( ) = PAPB ( ) ( A) + PAPB ( ) ( A) = 0.6 0.0+ 0.4 0.0 = 0.04. P( AB) PAPBA ( ) ( ) 0.6 0.0 PAB ( ) = = = = 0.4. P( B) PB ( ) 0.04 7.5 0 7.. A = {} B = {} P( BA ) P( B ). PBA ( ) =. 9 P( B) = P( A) P( B A) + P( A) P( B A) 7 = + =. 0 9 0 9 0 P( BA ) P( B). A B.
PBA ( ) = PB ( ) =. 0 0 P( BA ) = P( B). A B A B P( BA ) = P( B) P( AB) = P( A) P( B A) = PAPB ( ) ( ) A B P( AB) = P( A) P( B) A B. A B PA ( ) > 0 PB ( ) > 0 P( AB) = P( A) PB ( A) = PB ( ). A B A B A B A B. A B P( AB) = P( A) P( B). P( AB) = PA ( AB) = PA ( ) PAB ( ) = PA ( ) PAPB ( ) ( ) = P( A)[ P( B)] = P( A) P( B). A B. A B A B. A B A B A B A B.. 0.9 0.8. A = {} B = {} C ={}. PA= ( ) 0.9 PB ( ) = 0.8. C = AU B A B P( C) = P( AU B) = P( A) + P( B) P( AB) = P( A) + P( B) P( A) P( B) = 0.9 + 0.8 0.9 0.8 = 0.98.. A B C PAB ( ) = PAPB ( ) ( ) PBC ( ) = PBPC ( ) ( ) PCA ( ) = PCPA ( ) ( ) P( ABC) = P( A) P( B) P( C) 9
0 A B C. A, A, L, A k < k < < L < k P ( AA ) ( ) ( ) ( ) LA = PA k PA L PA k A, A, L, A. A, A, L, A k < k. A, A, L, A P( I A) = P( A) = = P( U A) P( A)..5. = = 0.8 0.70.5 A = { } =,,. P( AA A ) = P( AA A ) = P( AAA) = P( A) P( A ) P( A ) = 0. 0. 0.5 = 0.97. =.6 E A A E. A m P ( ) m. A A P( A) = p PA ( ) = p= q 0< p < A m ( ) m m m! m m P m = C p q = p q 0m..6. m!( m)! 0.8 0. 5 8. 0 5 5 5 5 P (5) = C 0.8 0. 0.75 0 0
8 Pm ( 8) = P (9) P (0) 0 0 = C 0.8 0. 0.8 0.9. 9 9 0 0. 0.6 0.4.. A :0 A :. PA = = PA ( ) = C 0.6 0.4 0.6 = 0.88. ( ) 0.6 0.6 PA ( + A) = PA ( ) + PA ( ) = 0.6 + 0.88 = 0.648. B :0 B : B :. PB = = PB ( ) = C 0.6 0.4 0.6 0.59 ( ) 0.6 0.6 4 PB ( ) = C 0.6 0.4 0.6 0.07. PB ( + B + B) = PB ( ) + PB ( ) + PB ( ) = 0.6 + 0.59 + 0.07 0.68. A = {} B = {} C = { } 4 5 6 7 8 9 0. A B C. C
0% A B 0% A C B C 8%5% %. A B C. 8. 4. 5 0. 6 0 0. 7. 7 0 4.. 4 8. 9 0 0 9 7 4.. 0 7.. 00 0.6 50 00. 00 0.. 4 5. α β.
6 0 4.. 7 0 0.8 5 0.4. 0 5 8 0.9 0.9. 0.86.. 9. 98 90 0... 0 0 5 5. 7 5... 0.5 0. 0.. 0.5 0. 0.08... 0.80.7 0.9. 0.8 0.4 0... 9... 4 4.. 0.8. 5 40%8%% 0.040.00.05..
4 6 /5//4 7 m. m k 8 0.90.8 0.85.. 9 0.00.00.050.04. 0 0.004 00. 80% 0 6. 0.8 0. 5 8. 0. 0 0.. 4 0.6. 99% 5 0 0 0.4 9 0. 8 0. 48. 6 0 0 50 0 0 7 0.40.5
0.7. 0. 0.6. 8 0.70 0.0 0.80 0.0.. α β θ. 5