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

Question

2. 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 (b) What is the long run proportion of the dice sitting at the lower left (c) As time t 00, does the position of the stone have a limiting distri- to the top right corner after one toss of the dice? corning? bution? If so, what is it?

Explanation / Answer

a)

Let the square be like this

4 is the lower left corner initial position

2 is the top right corner.

For every throw of the die, the movement will be as follows

For throws 2 and 6 the stone will move to position 2.

probability it will move to top right (2) with one throw = 1/6+1/6 =1/3

b)

To be at the lower left corner(4) with initial position of 4, the probabilities will be as follows after summarising the above table.

hence the long run proportion of the dice sitting at the left corner is 1/6

c)

As time reaches infinity, the probability distribution of each of the position is as follows.

Dice throw Probability Move to Position 1 1/6 1 2 1/6 2 3 1/6 3 4 1/6 4 5 1/6 1 6 1/6 2

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