Question

We have 3 red and 3 blue balls distributed to two boxes. Each box have equal number of balls (i.e. 3 balls). System is defined as ”in state i, i = 0, 1, 2, 3” if the first box contains i red balls. At each time step, a ball is drawn from the first box, another ball is drawn from the second box. The first ball is placed into the second box, and the second ball is placed into the first box. So the number of balls at any box stays constant (=3) at each time step n. Let Xn denote the state of the system after the nth step. Explain that this is a Markov Chain and calculate the transition probability matrix

Answer 1

This is a Markov Chain where the state of the system after each time step depends only on the number of red balls in the first box. The transition probability matrix can be calculated by determining the probabilities of transitioning from one state to another based on the number of ways the balls can be drawn from the boxes.

Step-by-step explanation:

By considering the system of boxes and balls described, we can see that this is a Markov Chain. A Markov Chain is a mathematical model that describes a system whose future state depends only on its current state and has no memory of its previous states. In this case, the state of the system after each time step depends only on the number of red balls in the first box, not on any previous states.

The transition probability matrix for this Markov Chain can be calculated by determining the probabilities of transitioning from one state to another. Since each ball has an equal chance of being drawn from its box, the probability of transitioning from state i to state j is equal to the number of ways in which i red balls can be drawn from the first box and j red balls can be drawn from the second box, divided by the total number of ways in which 3 red and 3 blue balls can be distributed among the boxes.

For example, to calculate the probability of transitioning from state 0 to state 1, we need to determine the number of ways in which 0 red balls can be drawn from the first box and 1 red ball can be drawn from the second box. This can be calculated using the combination formula: C(3,0) * C(3,1) = 1 * 3 = 3. The total number of ways in which 3 red and 3 blue balls can be distributed among the boxes is C(6,3) = 20. Therefore, the probability of transitioning from state 0 to state 1 is 3/20.