A Bayes belief net consisting of six nodes and its associated conditional probability tables are shown below.

P(a1)	P(a2)	P(a3)
0.5	0.3	0.2

	P(b1|ai)	P(b2|ai)
a1	0.4	0.6
a2	0.3	0.7
a3	0.5	0.5

P(c1)	P(c2)	P(c3)
0.2	0.4	0.4

	P(d1|bi,cj)	P(d2|bi,cj)
b1,c1	0.3	0.7
b1,c2	0.5	0.5
b1,c3	0.9	0.1
b2,c1	1.0	0.0
b2,c2	0.4	0.6
b2,c3	0.7	0.3

	P(e1|di)	P(e2|di)
d1	0.1	0.9
d2	0.8	0.2

	P(f1|di)	P(f2|di)	P(f3|di)
d1	0.1	0.5	0.4
d2	0.8	0.0	0.2

Problem:

a) Compute the probability P(a₃,b₂,c₃,d₁,e₂,f₁).

 

b) Compute the probability P(a₂,b₂,c₂,d₂,e₁,f₂).

 

c) Suppose we know the net is in the following (partial) state of evidence e: a₃,b₁,c₂. What is the probability P(f₁|e)? What is the probability P(e₂|e)?

 

d) Suppose we know the net is in the following (partial) state of evidence e: f₁,e₂,a₂. What is the probability P(d₁|e)? What is the probability P(e₂|e)?

 

e) Suppose we know the net is in the following (partial) state of evidence e: a₂,c₃,e₁,f₂. What is the probability P(d₂|e)? What is the probability P(d₂|e)?