from IPython. core.interactiveshell import InteractiveShell

InteractiveShell.ast$\_$node$\_$interactivity $=$ "all"

import numpy as np

import pandas as pd

np$.$set$\_$printoptions$($formatter$=\{\text{'}$float$\text{'}$: lambda x: $"\{0$:$0\mathrm{.}4$f$\}".$format$($x$)\})$

import random

import seaborn as sn

import matplotlib.pyplot as plt

import plotly.express as px

from numpy.linalg import matrix$\_$power

from numpy.linalg import linalg, inv

Problem $1$

Let the state transition matrix be defined as shown in the code cell below. Note that the row headings should be $(1,2,3)$ from top to bottom and the

column headings should be $(1,2,3)$ frm left to right.

a$.$ Find $P(x2=3|x0=1)=$

b$.$ Find $P(x3=2|x0=2)=$

c$.$ Find $P(x8=1|x0=3)=$

d$.$ Find $P(x10=1|x0=1)=$

e$.$ Find $P(x20=3|x0=2)=$

f$.$ Find $P(x200=2|x0=1)=$

g$.$ Find $P(x200=2|x0=2)=$

h$.$ Find $P(x200=2|x0=3)=$