Final answer:
A Markov chain for this scenario involves defining states based on ball distribution in boxes, calculating the transition probabilities between these states, and finding the transition probability matrix. The long-term probability of having at least 1 blue and 1 red ball in box 1 is found by summing appropriate steady-state probabilities.
Step-by-step explanation:
The student's schoolwork question involves setting up a Markov chain and finding the transition probability matrix for the problem of randomly distributing 2 red, 2 blue, and 2 yellow balls into two boxes, each with 3 balls, and then swapping one ball from each box at each time step. To answer this question, first define the states of the Markov chain to represent the different possible distributions of balls across the two boxes. For example, 'RBY' could represent one red, one blue, and one yellow ball in box 1. After defining the states, determine the transition probabilities between states (the probability of moving from one state to another in one time step) and organize them into a matrix called the transition probability matrix. Finding the long-run probabilities requires calculating the steady-state probabilities of the Markov chain. The probabilities in the long run that there is at least 1 blue ball and 1 red ball in box 1 can be computed by summing the steady-state probabilities of all states that satisfy this condition.