Please answer all parts and show clear working!

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 lim PIX, is a multiple of 11]. n-too Hint: define a suitable Markov chain with 11 states. (b) Define Yn = Xn + c where c is a constant integer and Xn 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 = j], where y; > 0 for j = 1, . ..,6, and _17; = 1. Does your answer to part (a) change? Give justification for your answer.Note: when answering (a), (b) and (c) above you should provide rigorous justification by appealing to results in Markov chain theory; just stating a correct answer will not be sufficient to gain full marks for that question part as many of the marks will be awarded for appropriate working and/or reasoning. Any results from Markov chain theory that you use should be stated (or referenced in the lecture notes, if you prefer) but should NOT be proved