Finding the conditional probability of 3 variables

Question

The question is:

P(Z,X)= 0.25

P (X)= 0.2

P(Y)= 1

P(Z|X)= 0.6

P(Z|Y)= 0.4

Find P(Z| X,Y).

I have 2 problems:

1. If P(Z|X)= 0.6 and using the conditional probability formula, shouldn't P(Z, X)= 0.12 but the probability given by the question is =0.2?

2. If I wish to use the conditional probability formula to find P(Z| X,Y), how do I go about finding P(X,Y,X) and P(X,Y)? Else, is there any other formulas/methods that I can use to find P(Z| X,Y)?

Answer 1

1. Yes: $\mathsf P(X)=0.2, \mathsf P(Z\mid X)=0.6,$ and $\mathsf P(X,Z)=0.25$ is inconsistent with the definition of conditional probability, which requires that: $\mathsf P(X,Z)=\mathsf P(X)\,\mathsf P(X\mid Z)$.   At least one from these values is erroneous.

2. Notice that $\mathsf P(Y)=1$ means that event $Y$ is certain.