If I roll a 6 sided die infinite times, each time calculating the sum of all rolls. What is the probability the sum will be 2023 after a certain roll.

Question

So after each time you roll the dice you find the sum of the current roll and the previous rolls. As this number get larger by 1 to 6 each time will it will pass 2023 at some point. What is the probability it will land exactly on 2023 after a roll?

I feel like I can solve this with some kind of a recursion or a matrix solution using a Markov Chain. However I am not sure how exactly to do this.

Answer 1

You'll want to solve the linear system of equations \begin{align*} p_n &= \tfrac16(p_{n+1}+p_{n+2}+p_{n+3}+p_{n+4}+p_{n+5}+p_{n+6}) \quad (0\le n\le 2022) \\ p_{2023} &= 1 \\ p_n &= 0 \quad (n\ge 2024). \end{align*} Do you see why this gives $p_0$ as the desired probability?