Some sample computations:

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<LI> Prob(D=T) <br>
P(D=T) = 
P(D=T,A=T,B=T) +
P(D=T,A=T,B=F) +
P(D=T,A=F,B=T) +
P(D=T,A=F,B=F) = <br>
P(D=T|A=T,B=T) P(A=T,B=T) +
P(D=T|A=T,B=F) P(A=T,B=F) +
P(D=T|A=F,B=T) P(A=F,B=T) +
P(D=T|A=F,B=F) P(A=F,B=F) = (since A and B are independent absolutely) <br>
P(D=T|A=T,B=T) P(A=T) P(B=T) +
P(D=T|A=T,B=F) P(A=T) P(B=F) +
P(D=T|A=F,B=T) P(A=F) P(B=T) +
P(D=T|A=F,B=F) P(A=F) P(B=F) = 
0.7*0.3*0.6 + 0.8*0.3*0.4 + 0.1*0.7*0.6 + 0.2*0.7*0.4 = 0.32
<LI> Prob(D=F,C=T) <br>
P(D=F,C=T) = 
P(D=F,C=T,A=T,B=T) +
P(D=F,C=T,A=T,B=F) +
P(D=F,C=T,A=F,B=T) +
P(D=F,C=T,A=F,B=F) = <br>
P(D=F,C=T|A=T,B=T) P(A=T,B=T) +
P(D=F,C=T|A=T,B=F) P(A=T,B=F) +
P(D=F,C=T|A=F,B=T) P(A=F,B=T) +
P(D=F,C=T|A=F,B=F) P(A=F,B=F) = (since C and D are independent given A and B)
P(D=F|A=T,B=T) P(C=T|A=T,B=T) P(A=T,B=T) +
P(D=F|A=T,B=F) P(C=T|A=T,B=F) P(A=T,B=F) +
P(D=F|A=F,B=T) P(C=T|A=F,B=T) P(A=F,B=T) +
P(D=F|A=F,B=F) P(C=T|A=F,B=F) P(A=F,B=F) = (since C is independent of B given A
and A and B are independent absolutely <br>
P(D=F|A=T,B=T) P(C=T|A=T) P(A=T) P(B=T) +
P(D=F|A=T,B=F) P(C=T|A=T) P(A=T) P(B=F) +
P(D=F|A=F,B=T) P(C=T|A=F) P(A=F) P(B=T) +
P(D=F|A=F,B=F) P(C=T|A=F) P(A=F) P(B=F) = 
0.3*0.8*0.3*0.6 + 0.2*0.8*0.3*0.4 + 0.9*0.4*0.7*0.6 + 0.8*0.4*0.7*0.4 = 0.3032


<LI> Prob(A=T|C=T) <br>
P(A=T|C=T) = P(C=T|A=T)P(A=T) / P(C=T). <br>
Now P(C=T) = P(C=T,A=T) + P(C=T,A=F) = 
P(C=T|A=T)P(A=T) + P(C=T|A=F)P(A=F)  = 0.8*0.3+ 0.4*0.7 = 0.52 <br>
So 
P(C=T|A=T)P(A=T) / P(C=T) = 0.8*0.3/0.52= 0.46.

<LI> Prob(A=T|D=F) <br>
P(A=T|D=F) = P(D=F|A=T) P(A=T)/P(D=F). 
Now P(D=F) = 1-P(D=T) = 0.68 from the first question above. <br>
P(D=F|A=T) = P(D=T,B=T|A=T) + P(D=F,B=F|A=T) =
P(D=F|B=T,A=T) P(B=T|A=T) + P(D=F|B=F,A=T) P(B=F|A=T) = (since B is independent 
of A)<br>
P(D=F|B=T,A=T) P(B=T) + P(D=F|B=F,A=T) P(B=F) = 0.3*0.6 + 0.2*0.4 = 0.26. <br>
So P(A=T|D=F) = P(D=F|A=T) P(A=T)/P(D=F) = 0.26 * 0.3 / 0.68 = 0.115


<LI> Prob(A=T,D=T|B=F). <br>
P(A=T,D=T|B=F) = P(D=T|A=T,B=F) P(A=T|B=F) = (since A and B are independent)
P(D=T|A=T,B=F) P(A=T) = 0.8*0.3 = 0.24. 
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