# Conditional Probability Calculation in Bayes Net

Say I have a simple Bayes Net that appears like that in the picture and am giving the following probabilities:

$$P(y|x) = 0.5$$

$$P(z|x)=0.4$$

$$P(y|\bar{x})=0.8$$

$$P(z|\bar{x}) = 0.9$$

How would I calculate the following, or is it not possible to calculate them? I think I need to know $$P(x)$$ to be able to calculate them:

$$P(y)$$

$$P(x|y \land z)$$

$$P(x|y)$$

## 1 Answer

How would I calculate the following, or is it not possible to calculate them? I think I need to know P(x) to be able to calculate them:

Yes.

$$P(y)=P(x)P(y\mid x)+P(\bar x)P(y\mid\bar x)$$

$$P(x\mid y,z) = \dfrac{P(x)P(y\mid x)P(z\mid x)}{P(x)P(y\mid x)P(z\mid x)+P(\bar x)P(y\mid\bar x)P(z\mid\bar x)}$$

$$P(x\mid y)=\dfrac{P(x)P(y\mid x)}{P(y)}$$