Consider a game in which a piece of stone is moved clockwise around the four ver

Question

Consider a game in which a piece of stone is moved clockwise around the four vertices of a unit square. At each time, a fair dice is tossed, and the stone is moved the number of steps as shown on the dice.

(a) Suppose the stone is initially at either the lower left or the lower right corner with equal probability. What is the probability that it move to the top right corner after one toss of the dice?

(b) What is the long run proportion of the dice sitting at the lower left corning?

(c) As time t , does the position of the stone have a limiting distri- bution? If so, what is it?

Explanation / Answer

The stone is initially at position 4.

This is the movement of the stone

probability of moving to the top right corner (position 2 ) = 1/6+1/6 = 2/6=0.33

b)

long run proportion of the dice sitting at the lower left corner (position 4)

= probability of position 4 = 1/6

c)

As time reaches infinity the probability distribution is as follows

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