Hi, could you please help me solve this

A player rolls a regular fair six-sided die repeatedly. Each time the die shows a 1, the player adds one point to his/her score. Each time the die shows a 4,5 or 6, the player removes one point from his/her score. If the die shows a 2 or a 3, the player's score does not change. The player begins with 8 points and plays repeatedly until his/her score first reaches 10 points (classed as a win for the player) or 0 points (classed as a loss for the player.) Let W. be the probability that the player wins overall, given that he/she has k points at a given time. i) Find a second order difference equation which must be satisfied by W. Clearly explain your reasoning. ii) Write down the boundary conditions, satisfied by W. and W,, Justify your answer. iii) Solve the difference equation to find W. . iv) Hence show that W =11.11% . Let / be the expected number of additional rolls required by the player until the game ends, given that he/she has k points at a given time. V) Find a second order difference equation which must be satisfied by N. Clearly explain your reasoning. vi) Write down the boundary conditions, satisfied by /, and N,. . Justify your answer. vii) Given that the solution of N. = A(3*) + B +3k , show that the largest number of expected additional games comes when the player has 8 points. The game is now modified so that only rolls of 1, 4, 5 or 6 are recorded. Rolls of 2 or 3 are no longer counted when adding up the number of rolls required. Let M, be the expected number of additional rolls now counted by the player until the game ends, given that he/she has k points at a given time. vili) Would the expected number of additional games required still be maximised when starting from 8 points? Justify your answer. ix) By what proportion would the expected number of rolls required change? i.e. would M. = - N. ? M. = -N. ? M. = -N,? etc. Justify your