Question

An urn initially contains 3 tags, of which 1 is red, 1 are green and 1 is blue. A tag is randomly selected from the urn and replaced with a tag of one of the others colors. For instance, if a red tag is selected, it will be replaced by either a blue or a green tag randomly (equally likely). This game continues until only tags of a single color remain in the urn.
a) What is the expected number of trials (games periods) in which we have exactly 2 colors in the urn?
b) What is the mean duration of the game?
c) What is the probability that the game ends with the urn containing only blue balls?

Answer 1

Let us consider that

Xn≤3, so that the set of states of the process is {0,1,2,3} being 0 an absorbing state. Now, since Xn+1=Xn or Xn+1=Xn−1, the probabilities you're interested in are P(3,3),P(3,2),P(2,2),P(2,1),P(1,1),P(1,0) (the others are 0 except for P(0,0)which is 1). These are going to be the elements of the matrix,

a)The propability for both of the tags you get are red or both green:

and the propability for one tag you get is red and one is green is :

c)The probability that the game ends with the urn containing only blue balls