What is the distribution of the following event?

[number0]

Homework Statement

See uploaded imagery.

Homework Equations

The Attempt at a Solution

See uploaded imagery.


I know that the distribution has something to do with the 1-dimensional Ising model being sampled via Gibbs sampler method, but that is all I know. Anyone have a hint/solution? Thanks.

 

[Ray Vickson]

For any $a, b, c \in \{-1,1\}$ we have
$$P(x=a|y=b,z=c) = \frac{P(x=a \cap \{y=b,z=c\})}{P(y=b,z=c)} = \frac{P(x=a,y=b,z=c)}{P(y=b,z=c)},$$
where $P(y=b,z=c) = \sum_{w=-1}^1 P(x=w,y=b,z=c).$ This has nothing to do with Physics or the Ising model or anything else like that; it is just elementary probability.

RGV

 

[number0]

Sorry, but I already got your answer (see imagery)... it's just that I do not know what distribution the conditional distribution follows...