# Inter-causal reasoning: How to solve probability with two conditions?

Below is the scheme of conditional dependence and the probabilities of events:

P(A=1) = 0.01
P(A=0) = 0.99
P(B=1) = 0.1
P(B=0) = 0.9
P(C=1|A=0,B=0) = 0.1
P(C=1|A=0,B=1) = 0.5
P(C=1|A=1,B=0) = 0.6
P(C=1|A=1,B=1) = 0.9


Given the probabilities above I wanted to calculate P(B=1|C=1) and P(B=1|C=1,A=1) but didn't get the correct result.

I wrote the probabilistic function the following way:

P(A, B, C) = P(A)P(B)P(C|A, B)


and then set the variables

P(B=1, C=1) = P(A=0, B=1, C=1) + P(A=1, B=1, C=1)=
=P(A=0)P(B=1)P(C=1|A=0, B=1) + P(A=1)P(B=1)P(C=1|A=1, B=1)=
=0.99*0.1*0.5 + 0.01*0.1*0.9 = 0.0495


The result however is not correct and don't know where is the error. I would be very thankful if anyone could correct/explain what's wrong.

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You said you were asked P(B=1|C=1) but you computed P(B=1,C=1) hence you are off by the factor P(C=1) (which you will need to compute). – Did Mar 30 '12 at 11:06
Is P(C=1) = ( P(C=1|A=0,B=0)+P(C=1|A=0,B=1)+P(C=1|A=1,B=0)+P(C=1|A=1,B=1) )/4 ? – Kaushik Acharya Apr 1 '12 at 11:53

The typical way I do inter-causal reasoning is to flip the conditional probabilities around --

P(B=1|C=1) = P(B=1,C=1) / P(C=1)
= P(C=1|B=1) P(B=1) / P(C=1)

P(B=1|C=1,A=1) = P(B=1,C=1,A=1) / P(C=1,A=1)
= P(C=1|B=1,A=1) P(B=1,A=1) / P(C=1,A=1)
= P(C=1|B=1,A=1) P(B=1) P(A=1) / P(C=1|A=1)P(A=1)
= P(C=1|B=1,A=1) P(B=1) / P(C=1|A=1)


Does that help?

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And how do you calculate P(C=1) ? Is it ( P(C=1|A=0,B=0) + P(C=1|A=0,B=1) ) * ( P(C=1|A=1,B=0) + P(C=1|A=1,B=1) ) ? – jaor Jul 30 '14 at 13:02