In this experiment there are two urns One contains 6 red bal

In this experiment there are two urns. One contains 6 red balls and the other 6 blue balls. An experiment consists of picking two random balls from each urn and putting them in the opposite urn. Using Markov Chains create a table containing the columns: round number and the probability that the experiment ends exactly at that round. Proceed till the sum of all the probabilities is higher than 0.90. Report the states. Then create Markov diagram and transition matrix. Finally use the transition matrix to create the table.

Solution

n this case, let\'s find the probability that we draw one ball of each color. There are eight total balls, so there are \"eight choose two\" or (82)=28(82)=28 distinct pairs that can be drawn. There are 53=1553=15 ways to pick one white and one black. So the probability of one of each is 15281528.    

Now it must be the case that either we get one of each color OR we get two of the same color. Since these two probabilities must add to one, that leaves 11528=132811528=1328 or just less than 50% as the probability of two of the same color.

If the balls are returned after being drawn, then there are 64 distinct (ordered) outcomes and 30 of them have balls of different color (any of the five white followed by any of the three black or vice versa = 532=30532=30). So the probability of two different balls is 3064=15323064=1532 leaving the probability of two matching balls to be 11532=173211532=1732 or just over 50%.  

It stands to reason that the probability of two matching balls should go up slightly if the balls are returned after being drawn since it is, in this case, possible to draw the same ball twice -- an event that was not possible when balls were not returned