In each part, give the one-step transition matrix P of the described time-homoge

Question

In each part, give the one-step transition matrix P of the described time-homogeneous Markov chain {X_n: n greaterthanorequalto 0}. Also, list the states to the immediate left side of the matrix. a) There are two balls and two compartments. Ball 1 is in compartment A and ball 2 is in compartment B. A ball is selected at random from the two balls and is moved to the compartment that it is not currently in. This process is repeated indefinitely. Let X_n be the number of balls in compartment A after the nth ball has been moved. To get full credit, explain how you found the transition probabilities. (+10) b) An um contains two red balls and one green ball. At each time n, a ball is selected at random from the three balls, removed, and replaced by a ball of the opposite color (i.e., a red ball is replaced by a green ball and a green ball is replaced by a red ball). This process is repeated until the three balls are all red or all green; once that happens, no more balls are selected. Let X_n be the number of red balls in the um at the end of time n. Note that states 0 and 3 are absorbing. To get full credit, explain how you found the transition probabilities. (+10)

Explanation / Answer

There are 2 compartments A and B and A has ball1 and B has ball 2.

Let us call this state as A1B2. There are 3 other different states possible, A2B1(ball 2 in A, ball 1 in B), A12B(both the balls are in A), AB12 (both the balls are in B) and A1B2 (

From the state A1B2

when ball 1 is selected it shifts to B, then the state becomes AB12 (where B compartment has 1 and 2 numbered balls)

when ball 2 is selected it shifts to A, then the state becomes A12B (where A compartment has 1 and 2 numbered balls)

Therefore the movement to both of these states have probability of 1/2 each. From A1B2, A2B1 cannot be reached and it cannot move to it's same state

Doing this for all the other states, we get

initial state is

One step transition matrix is

b) Urn contains 2 R and 1 G. Let us call this state as 2R1G.

Other states possible out of 3 balls of 2 colors are 3G, 3R, 2G1R and 2R1G(initial state)

From 2R1G, it can move to 3R if we pick green ball or can move to 1R2G if we pick red ball, it cannot go to it's own state or 3G. Since there are only possibilities, each probability is 1/2

Similarly From 2G1R, it can move to 3G if we pick RED ball or can move to 2R1G if we pick green ball, it cannot go to it's own state or 3R.

From the other 2 states movement is not possible as they are final states.

Initial state is

one step transition is

A1B2 AB12 A12B A2B1 A1B2 0    1/2 1/2 0    AB12 1/2 0    0    1/2 A12B 1/2 0    0    1/2 A2B1 0    1/2 1/2 0   

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