# Solving the marginal probability given the probability distributions

Given the probability distributions as follows

$$P(b_1|a_0,c_0)=p$$

$$P(b_1|a_1,c_0)=o$$

$$P(b_1|a_0,c_1)=n$$

$$P(b_1|a_1,c_1)=m$$

$$P(a_1|c_1)=x$$

$$P(a_1|c_0)=y$$

$$P(c_1)=r$$

I need to find $$\dfrac{P(b_1|a_0)}{P(a_0)}$$

My attempt:

$$\dfrac{P(b_1|a_0)}{P(a_0)}=\dfrac{P(b_1|a_0)}{P(a_0,b_0)+P(a_0,b_1)}=\dfrac{P(a_0,b_1,c_1)+P(a_0,b_1,c_0)}{P(a_0,b_0,c_0)+P(a_0,b_0,c_1)+P(a_0,b_1,c_0)+P(a_0,b_1,c_1)}$$

I was able to solve the numerator using the given probabilities and the formula $$P(a,b,c)=P(b|a,c)\cdot P(a|c)\cdot P(c)$$ but I cannot figure out how to solve the denominator part

• How are $x_1$ and $z_1$ related to $a,b,c$? Commented Sep 19, 2020 at 23:02

Although you haven't explicitly stated, it seems that there are three RVs $$a,b,c$$ with binary values, e.g. $$a_1$$ meaning $$a=1$$.
Given the last three probabilities, i.e. $$P(a_1|c_1), P(a_1|c_0), P(c_1)$$ you can find the complete joint distribution of $$P(a,c)$$. And, when you multiply the first four probabilities with corresponding joint distributions of $$a$$ and $$c$$, you can find the complete joint distribution of $$P(a,b,c)$$, from which you can find any probability you like.
For example: $$P(a_0,b_0,c_0)=P(b_0|a_0,c_0)P(a_0,c_0)=(1-p)P(a_0|c_0)P(c_0)=(1-p)(1-y)(1-r)$$
Besides, your numerator is incorrect, it should have been the following: $$P(b_1|a_0)=P(c_1,b_1|a_0)+P(c_0,b_1|a_0)$$