Here is the problem: $A$ and $B$ roll a dice taking turns with $A$ starting this process. Whichever one rolls the first $6$, wins. Find the probability of $A$ winning.
I know how to solve this problem using a geometric sequence summation. I came across this problem which said "Hint:you will have to use a geometric series to find the answer".
I began to wonder if there was an alternative solution which doesn't use geometric series. I am not sure where to begin even. Does anyone know of such a solution? I am talking about an analytical solution not a simulation or a monte carlo method of doing it.