Question

A number x is selected at random in the interval [-3,3]. Let the events A={x < 0}, B={|x - 0.5| < 0.5|}, and C={x > 0.75}. Find P[A|B], P[B|C],P[A|C^C],P[B|C^C].

Answer 1

from above B ={-0.5 <x-0.5 <0.5 } ={0 <X<1)

P(A) =P(X<0) =(0-(-3))/(3-(-3))=1/2

P(B) =(1-0)/(3-(-3))=1/6

P(C) =(3-0.75)/(3-(-3))=2.25/6 =3/8

P(B n C) =P(0.75 <X<1 )=(1-0.75)/(3-(-3)) =0.25/6 =1/24

P(A n C^c) =P(X<0) =P(A) =1/2

P(B n C^c) =P(0 <x<0.75) =(0.75-0)/(3-(-3))=0.75/6 =1/8

(A n B) ={ }

therefore P(A|B )=P(A n B)/P(B) =0/(1/6)=0

P(B|C) =P(B n C)/P(C) =(1/24)/(3/8)=1/9

P(A | C^c) =P(A n C^c)/P(C^c) =P(A)/P(C^c) =(1/2)/(1-3/8) =4/5

P(B |C^c) =P(B n C^c)P(C^c) =(1/8)/(1-3/8)=1/5