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 Xn 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 Hint: define a suitable Markov chain with 11 states. (b) Define Yn lim P[X, is a multiple of 11]. n = Xn + c where c is a constant integer and Xn is defined as before. Find lim P[Y, is a multiple of 11]. n j], where yj > 0 (c) Suppose now that the die is not necessarily fair and that 7; = P[X 6, and =17; 1. Does your answer to part (a) change? Give = for j 1, justification for your answer. = =