Please help me solve part (b) of this problem. There are two parts (i) and (ii). Thanks.

(a) Consider the following scenario: suppose the probability that a burglar breaks into your car is m, and the probability that an innocent passerby accidentally touches your car is 7n. Let Z, be a binary random variable that is 1 if there is a burglar, and 0 otherwise. Likewise, let Zi be a binary random variable that is 1 if there is an innocent passerby, and 0 otherwise. Suppose Z, and Zi are independent of each other. Let X be a binary random variable that is 1 if your car alarm goes off. The probability your car alarm goes off depends on Zb and Zi, and is known to be: "El- (b) Suppose you know the parameters 7Tb and 7m, as well as P(X = 1 | Z5, Z1) as specied in Part (a). X is the observed variable, and Z and 2;, are the latent (unobserved) variables. We want to sample from lP'(Z, 2;, | X, 717;, 7a), the posterior over the latent variables conditioned on everything else. We'll use Gibbs sampling to do this: (i) Suppose we are running Gibbs sampling, and on each iteration we sample Zo first and then sample Zi. We observed X = 0, and the values of Zo and Zi from iteration t are zit) = 0 and Z,) = 1. Derive the distribution used for the Gibbs sampling update of Z.". Your solution should be in terms of Tb, Ti, and constants. (ii) Now, suppose we draw Z, 7(t+) = 1 from the distribution derived in Part (b.i). Derive the distribution used for the Gibbs sampling update of Z.". Your solution should be in terms of Tb, Ti, and constants