Question

Suppose X and Y are conditional independent given Z and the following probabilities P(X,Y|Z), what is the joint probability? P(Z)=0.5 P(X | Z)=0.5 P(Y | Z)=0.4

Answer 1

0.2

Explanation:

To find the joint probability P(X, Y | Z), we can use the definition of conditional independence:

If X and Y are conditionally independent given Z, then P(X, Y | Z) = P(X | Z) * P(Y | Z).

Given the probabilities provided:

P(Z) = 0.5

P(X | Z) = 0.5

P(Y | Z) = 0.4

Using the formula for conditional independence:

P(X, Y | Z) = P(X | Z) * P(Y | Z) = 0.5 * 0.4 = 0.2

So, the joint probability P(X, Y | Z) is 0.2.