Question 3. [10 Marks] A die with sides labelled by the numbers 1 to 6 is thrown repeat- edly, with the outcome of each throw being independent of the other throws. Let X, denote the sum of the first n throws. (a) Assuming the die is fair, so that the six numbers are equally likely in a given throw, find lim PIX, is a multiple of 11]. n-00 Hint: define a suitable Markov chain with 11 states. (b) Define Yn = Xn + c where c is a constant integer and X, is defined as before. Find lim P[Yn is a multiple of 11]. n-+00 (c) Suppose now that the die is not necessarily fair and that y; = P[X1 = ], where y; > 0 for j = 1, ..., 6, and _;= = 1. Does your answer to part (a) change? Give justification for your