Consider the Dishonest Casino example from the Rcode page for HMM: a dishonest casino uses two dice, one of them is fair and the other is loaded.

• The probabilities of the fair die are (1/6, ..., 1/6).
• The probabilities of the loaded die are (1/10, ..., 1/10,1/2)

LetX_t

Xt

denote the observed outcome at time t, which takes values from 1 to 6. LetZ_t=1

Zt

=1denote that a fair die is used at timet

t, andZ_t=2

Zt

=2denote that a loaded die is used at timet

t.

We modelZ_t

Zt

's by a Markov chain. Assume the initial distribution forZ_1

Z1

is (0.5, 0.5), andP(Z_{t+1} = k | Z_t = k) = 0.6

P(Zt+1

=kZt

=k)=0.6fork=1, 2

k=1,2.

Compute the following probabilities. Express your answers as fractions reduced to the lowest terms.