What Are the Markov Chain Probabilities for a Taxicab's Moves?

[lina29]

Homework Statement

A taxicab moves between the airport, Hotel A, and Hotel B according to a Markov chain with transition probabilities:
P(airport → A) = 0.7,
P(airport → B) = 0.3,
P(A → airport) = 0.9,
P(A → B) = 0.1,
P(B → airport) = 0.8,
P(B → A) = 0.2.
A-If the taxicab starts at the airport, what is the probability that it will be at Hotel A two moves later?
B-Suppose the taxicab starts at the airport with probability 0.6 and starts at Hotel A and Hotel B with probability 0.2 each. What is the probability that it will be at Hotel B two moves later?
C- In the long run, what fraction of visits will the taxicab make to each of the three locations?

Homework Equations

The Attempt at a Solution

I have gotten the answer for parts A & C but I don't understand at all how I would set up the matrix with part B. My initial though was that the matrix was

0 .2 .8
.2 0 .8
.6 .4 0

with the first row/column being hotel A, the second row/column being hotel B, and the third row/column being the airport

then by squaring the matrix I got
.52 .32 .16
.48 .36 .16
.08 .12 .8

and then to get B I added up .32 + .12=.44 which was wrong. What did I do wrong?

 

[Ray Vickson]

For (b): in two moves, P{at B} = P{at B|start at port}*P{start at port} + P{at B|start at A}*P{start at A} + P{at B|start at B}*P{start at B}. The conditional probabilities are given by P*P.

RGV