Q1
P(A∩C∣B)=P(B)P(A∩C∩B)=P(B)P(A)P(B∣A)P(C∣B)=P(B)P(B∣A)P(A)P(C∣B)=P(A∣B)P(C∣B)
Q2
Please find attached at the end of this document
Q3
a
P(A∩B∩C)=P(C∣A∩B)P(A∩B)=0.2×P(B∣A)P(A)=0.2×0.75×0.3=0.045
b
P(Bc∩C)=P(Bc∩C∩A)+P(Bc∩C∩Ac)=P(C∣A∩Bc)P(A∩Bc)+P(C∣Ac∩Bc)P(Ac∩Bc)=0.8×P(Bc∣A)P(A)+0.9×P(Bc∣Ac)P(Ac)=0.8×(1−0.75)×0.3+0.9×(1−0.2)×(1−0.3)=0.564
c
P(Bc∩C)P(C)=P(C)−P(B∩C)=P(Bc∩C)+P(B∩C)=0.564+P(B∩C∩A)+P(B∩C∩Ac)=0.564+0.045+P(C∣Ac∩B)P(Ac∩B)=0.564+0.045+0.15×P(B∣Ac)P(Ac)=0.564+0.045+0.15×0.2×(1−0.3)=0.63
d
P(A∣C∩Bc)=P(C∩Bc)P(C∩Bc∣A)P(A)=P(C∩Bc)P(A)P(C∩Bc∩A)P(A)=0.564P(C∩Bc∩A)=0.564P(C∣A∩Bc)P(A∩Bc)=0.5640.8×P(Bc∣A)P(A)=0.5640.8×(1−0.75)×0.3=0.106382978