Final answer:
The transition matrix P for Mark Hovchain's ball selection Markov chain is created by calculating the transition probabilities based on either selecting a red ball (state stays the same) or a blue ball (state advances to the next, and the blue ball is replaced with a red one).
Step-by-step explanation:
We are tasked with creating the transition matrix, P, for a Markov chain that represents the process taking place with Mark Hovchain's box of red and blue balls. In this Markov chain, the states correspond to the number of red balls currently in the box. We will denote the states as R5, R6, R7, and R8 to represent the box containing 5, 6, 7, or 8 red balls, respectively.
When a red ball is drawn, the number of red balls in the box remains unchanged, since the red ball is replaced. However, drawing a blue ball changes the state because the blue ball is removed, and an additional red ball is put into the box. Therefore, the probability of moving from state Ri to state Ri+1 (where i is 5, 6, or 7) is the probability of drawing a blue ball when there are i red balls and 8 - i blue balls in the box.
The transition probabilities are as follows:
- P(R5 to R5) is the probability of picking a red ball when there are 5 red and 3 blue balls, which is 5/8.
- P(R5 to R6) is the probability of picking a blue ball in the same condition, which is 3/8.
- P(R6 to R6) is 6/8, since there's now one less blue ball.
- P(R6 to R7) is 2/8.
- P(R7 to R7) is 7/8, with only one blue ball left.
- P(R7 to R8) is 1/8.
- States R8 and beyond remain absorption states since there are only red balls left to draw, so P(R8 to R8) is 1.
Therefore, the transition matrix P for this Markov chain is:
R5R6R7R8R55/83/800R606/82/80R7007/81/8R80001
Note that all other transitions have a probability of 0 since the specified transitions are the only possible ones given the rules of the box's ball replacement.