Question 3. [10 Marks] A die with sides labelled by the numbers 1 to 6 is...

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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[X₁ 6, and Σ=17; 1. Does your answer to part (a) change? Give = for j 1, justification for your answer. = = 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[X₁ 6, and Σ=17; 1. Does your answer to part (a) change? Give = for j 1, justification for your answer. = =

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3a lim PX is a multiple of 11 If the die is fair then the probability of each of the numbers 1 to 6 ... View the full answer