We first examine a simple hidden markov model (hmm). we observe a sequence of rolls of a four-sided die at an "occasionally dishonest casino", where at time t the observed outcome x_t ∈{1, 2, 3, 4}. at each of these times, the casino can be in one of two states z_t ∈ {1.2). when z_t = 1 the casino uses fair die, while when z_t=2 the die is biased so that rolling a 1 is more likely. in particular: p(x_t = 1| z_t = 1) = p(x_t = 2| z_t = 1) = p(x_t = 3 | z_t = 1) = p(x_t = 4|z_t = 1) = 0.25, p(x_t = 1| z_t = 2) = 0.7, p(x_t = 2 | z_t = 2) = p(x_t = 3 | z_t = 2) = p(x_t = 4 | z_t = 2) = 0.1. assume that the casino has an equal probability of starting in either state at time t = 1, so that p(z1 = 1) = p(z_1 = 2) = 0.5. the casino usually uses the same die for multiple iterations but occasionally switches states according to the following probabilities p(z_t + 1 = 1 | z_t = 1) = 0.8, p(z_t + 1 = 2) = 0.9 the other transition probabilities you will need are the complements of these. hint: for all of the question below, you should not need to enumerate all possible state sequences. a. under the hmm generative model, what is p(z1 = z2 = z3), the probability that the same die is used for the first three rolls? b. suppose that observe the first two rolls. what is p(z1 = 1 | x1 = 2, x2 = 4), the probability that the casino used the fair die in the first roll?