Steady state vector and probability after 3 transitions in Markov chain

[mr_coffee]

Hello everyone, confused. the directions to this problem are the following:
Find the steay-steat vector, and assuming the chain starts at 1, find the probability that it is in state 2, after 3 transitions.
well i got the problem and i got the S0 to S3, because it said after 3 transitions, is that what they wanted ,or did they want the lorn term steady state vector? also how do i find the probabliy?> Thanks.
Picture is here:
http://show.imagehosting.us/show/758415/0/nouser_758/T0_-1_758415.jpg

 

[member 553083]

The steady-state vector is the long-term probability distribution of the Markov chain, which tells you the probability that the chain will be in each state after a large number of transitions. To find the steady-state vector, you need to solve the system of linear equations: pi * P = piwhere pi is the steady-state vector and P is the transition matrix. In your case, the transition matrix is: P = [0.6 0.4; 0.5 0.5]and the steady-state vector should satisfy the equation: [p1; p2] * [0.6 0.4; 0.5 0.5] = [p1; p2]Solve this equation to find the steady-state vector.To find the probability that the chain is in state 2 after 3 transitions, you can use the transition matrix. The probability that the chain is in state 2 after 3 transitions is equal to the element in the second row and third column of P^3 (where P^3 is the cube of the transition matrix). In your case, P^3 = [0.51 0.49; 0.51 0.49], so the probability that the chain is in state 2 after 3 transitions is 0.49.