Markov chain?

Question

1)Paul travels back and forth in his work between New York and Boston. He never makes two trips on the same day. On a day when he wakes up in New York, the probability is 0.04 that he will fly to Boston during the day. On a day when he wakes up in Boston, the probability is 0.12 that he will fly to New York.
If he wakes up in Boston on Monday, what is the probability he will wake up in New York on Wednesday?
If he wakes up in New York on Monday, what is the probability he will wake up in New York on Wednesday?
If he wakes up in New York on Monday, what is the probability he will wake up in Boston on Thursday?
If he wakes up in Boston on Monday, what is the probability he will wake up in Boston on Thursday?
: I tried..but can't still find the right answer so please explain..
2) Ed, Bob, and Peg have responsibility for dishwashing in their home. If Bob washes dishes one day, Peg does it the next day. If Peg washes dishes, then Ed washes dishes the next day. When Ed washes dishes, there is a 20% chance that Peg will wash dishes the next day and a 80% chance that Bob will. If T is the transition matrix for this problem, what is the smallest value of n for which the matrix Tn has no zero entries?


In the long run what fraction of the time do each of these people wash dishes? (Be accurate to 4 decimal places.)

Ed?
Bob?
Peg?

Answer 1

problem 1
If he is in Boston on Monday, to be in NY on Wed
there are these possiblities

Stay in Boston on Mon, (1-.12)
and Travel to NY in Tu (0.12
or
Travel to NY on Mon (0.12)
and Stay in NY on Tue (1-0.04)
So (1-.12)(.12) + (.12) (1-.04) = .22

Use similar logic to find the rest. It may help to draw a tree
with all possible combinations.
CityA
A B
A B A B
A B A B A B A B