Conditional probability PDF Now let us ask

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?? ? ?????? ????? ???? ??? ?????? ?? ? ?????? ??? ????? ?????? ??? ?????? ?? ??20=100

P(EjF) :=P(E\F)P(F):????P(E\F) =P(EjF)P(F)?

???? ??? ???? ??? ????? ? ?? ???B?? ??? ????? ???? ??? ???? ??? ????? ? ?? ????P(A\B)

P(AjB) =1=361=6= 1=6?

???? ?? ????? ? ??? ??? ?? ? ??? ? ????? ??? ?? ?? ????P(A\B) =3 2 2 0=5

2? ??? ??

P(AjB) =3=103=5= 1=2?

???P(B) =P(bb;bg;gb) = 3=4? ?? ??? ?????? ??1=43=4= 1=3? P(DjT) =P(D\T)P(T)=(0:99)(0:003)(0:99)(0:003) + (0:01)(0:997)23%: ??????? ??? ????P(EjF)??? ??? ???? ?? ???P(FjE)? ?????? ????

P(E\F) =P(EjF)P(F);

?????P(D) = 0:36;P(C) = 0:30;P(CjD) = 0:22?? ???? ?? ????P(DjC)? ?? ???? (0:22)(0:36) = 0:0792? ??P(DjC) = 0:0792=0:3 = 0:264 = 26:4%? ??? ????? ?? ????? ? ???? ?? ??? ?????P(AjW) = 0:30;P(AjM) = 0:25;P(W) = 0:60

P(WjA) =P(W\A)P(A):

P(W\A) =P(AjW)P(W) = (0:30)(0:60) = 0:18:

?? ???P(A)? ?? ?????

P(A) =P(W\A) +P(M\A):

P(M\A) =P(AjM)P(M) = (0:25)(0:40) = 0:10:

P(A) =P(W\A) +P(M\A) = 0:18 + 0:10 = 0:28:

??P(E)>0? ????

P(E) =P((E\F)[(E\Fc)) =P(E\F) +P(E\Fc):

P(FjE) =P(E\F)P(E)=P(EjF)P(F)P(E\F) +P(E\Fc)

P(EjF)P(F)P(EjF)P(F) +P(EjFc)P(Fc):

??????? ??? ??? ????? ? ?? ??? ????P(Njn) = 0???P(NjN) = 1? ??? ??????? ?? ?? n??????? ?? ??? ??????? ??? ???? ????? ???? ????? ??? ?????? ???? ??????n+ 1??n1 y(n) =12 y(n+ 1) +12 y(n1): y(n+ 1)y(n) =y(n)y(n1): y(n) =y(n)y(0) = (y(n)y(n1)) +:::+ (y(1)y(0))??????? =n(y(1)y(0)) =ny(1): ???? ?? ? ??????? ???z(n)? z(n) =12 z(n+ 1) +12 z(n1): z(n)z(0) = (z(n)z(n1)) +:::+ (z(1)z(0)) =n(z(1)z(0)); z(n) =n(z(1)z(0)) +z(0): z(n) =an+b: z(1) =z(0); z(n) =z(0) z(n) = 1??? ???n:

UMc??? ?????U\Mc=U? ?????

P(UjMc) =P(U\Mc)P(Mc)=P(U)1P(M)=4=106=10=23

???? ??? ??52 248

11???? ???? ??????? ??? ???? ?? ????

P(A) =

4 248
11 52
13 1339
??4 248
112
137

12??? ?????

P(A\B) =

4 248
112
137
12 52
1339
13

P(BjA) =P(A\B)P(A)=

4 248
112
137
12.52 1339
13 4 248
11.52

13=237

12 39
13

P(??????? ?????<$25K) =212500

+36500
=248500 P(???? ?????>$25Kj??????? ?????>$25K) =54=500(198 + 54)=500=54252 = 0:214 P(???? ?????>$25Kj??????? ?????<$25K) =36=500248=500= 0:145: ???E;F2 F?? ?????? ????P(E);P(F)>0? ???? ???P(E\F) =P(E)P(FjE)? ????P(E) =P(EjF)P(F) +P(EjFc)P(Fc)? ????E\F???E\Fc? ?????(i)?? ????

P(E) =P(E\F) +P(E\Fc)

=P(EjF)P(F) +P(EjFc)P(Fc): 45
17

P(A\C) =P(C)P(AjC) =12

17 =114 ???E1;E2;:::;En2 F?? ??????? ????

P(E1\E2\:::\En)

P(B1\B2\B3) =P(B1)P(B2jB1)P(B3jB1\B2) =513

614
715
P(???? ? ???? ????) =P(B1\R2\R3) +P(R1\B2\R3) +P(R1\R2\B3) = 3589131415: i=1Fi? ????? ??? ??? ?????E2 F?? ????? ????

P(E) =nX

P(A1) =P(A1jA)P(A) +P(A1jAc)(1P(A))

= 0:40:3 + 0:2(10:3) = 0:26 P(AjA1) =P(A\A1)P(A1)=P(A)P(A1jA)0:26=0:30:40:26=614 i=1Fi? ????? ??? ??? ?????ES??? ???j= 1;:::;n?? ????? ????

P(FjjE) =P(EjFj)P(Fj)P

P(D) =P(I)P(DjI) +P(II)P(DjII) +P(III)P(DjIII)

= 0:350:02 + 0:250:01 + 0:40:03 =21510;000: P(IIIjD) =P(III)P(DjIII)P(D)=0:40:03215=10;000=120215 P(KjC) =P(CjK)P(K)P(CjK)P(K) +P(CjKc)P(Kc)=1p1p+1m (1p)=mp1 + (m1)p: B=f??? ?? ??? ???? ?????? ?g? ???C=f?? ????? ??? ?? ??? ???? ????? ? ?g? ???? ?? P(AjC)? ???? ?? P(BjC)? ???? ??? ??? ?? ?? ?????9????? ???? ??? ?????? ?? ????? ???6? P

E(EcjF) = 1E(EjF):

A=f(1;2);(2;1)g

B=f(1;6);(2;5);(3;4);(4;3);(5;2);(6;1)g

P(AjC) =P(A\C)P(C)=2=3611=36=211

P(BjC) =P(B\C)P(C)=2=3611=36=211

F??? ????? ??????? ??? ?? ?? ????? ?? ?? ???? ?? ???P(FjE)? ????? ?? ?????? ???? f(6;3);(6;4);(6;5);(6;6);(3;6);(4;6);(5;6)g? ??P(E\F) =736

P(FjE) =P(E\F)P(E)=7=3611=36=711

??? ??????? ???????P(B1) =P(B2) =12

P(GjB2) =25

P(B1jG) =P(GjB1)P(B1)P(GjB1)P(B1) +P(GjB2)P(B2)=1=41=4 + 1=5=59 ?????? ??E??? ????? ????? ??????? ?? ??????? ?? ??? ??? ??? ??F??? ????? ????? ??????? ???? ??? ??? ?? ???? ????P(E) = 0:6???P(FjE) = 0:15? ???? ????? ????P(E\F) = (0:6)(0:15) = 0:09? ??? ?? ?? ????? ????P(Ec\F) = 0? ?????P(F) =P(E\F)+P(Ec\F)? ?? ??? ????P(F) = 0:09? ????? ???E?? ??? ????? ????? ??????? ?? ??????? ?? ??? ??? ??? ???F?? ??? ????? ????? ??????? ???? ??? ??? ?? ???? ????P(E) = 0:6???P(FjE) = 0:8? ???? ????? ????P(E\F) = (0:6)(0:8) = 0:48? ??? ?? ?? ????? ????P(Ec\F) = 0? ?????P(F) =

P(E\F) +P(Ec\F?? ?? ??? ????P(F) = 0:48?

P(MjC) =P(CjM)P(M)P(CjM)P(M) +P(CjMc)P(Mc)

(0:05)(0:49)(0:05)(0:49) + (0:0025)(0:51)0:9505: ????? ????? ???? ????? ???? ???????X? ???Y?? ??? ?????? ???? ????? ???? ???????Y?

P(H) =P(HjX)P(X) +P(HjY)P(Y)

= (0:99)(0:6) + (0:95)(0:4) = 0:974:

P(YjH) =P(HjY)P(Y)P(H)

(0:95)(0:4)0:9740:39:

P(XjHc) =P(HcjX)P(X)P(Hc)=P(HcjX)P(X)1P(H)

(10:99)(0:6)10:974=(0:01)(0:6)0:026 0:23 ? ???? ?? ???? ???????A? ??B??? ????? ????? ???? ?? ???? ???????C? ???? ?? ?????? P(AjD) =P(DjA)P(A)P(DjA)P(A) +P(DjB)P(B) +P(DjC)P(C) (0:05)(0:25)(0:05)(0:25) + (0:04)(0:35) + (0:02)(0:4)= 0:362:

P(KjC) =P(CjK)P(K)P(C)=

P(CjK)P(K)P(CjK)P(K) +P(CjE)P(E) +P(CjG)P(G)=

112
112
+13 14 +14 14 =2431 :774;

P(Dj+) =P(+jD)P(D)P(+jD)P(D) +P(+jDc)P(Dc)

(0:95)(0:01)(0:95)(0:01) + (0:005)(0:99)= 0:657: P

F(S) =P(SjF) =P(S\F)P(F)=P(F)P(F)= 1:

??? ????fEig1 P F 1[ i=1E i! =P 1[ i=1E ijF! P((S1 i=1Ei)\F)P(F) P(S1 i=1(Ei\F))P(F)=P 1 i=1P(Ei\F)P(F) 1X i=1P(Ei\F)P(F)=1X i=1P

F(Ei):

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[PDF] Conditional probability Now let us ask