ALternate solution to a probability problem

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.

• It appears that the major step in any solution is to use conditional probability. If both Player A and Player B fail to roll a 6 on a given turn, then you're back at square 1, as far solving the problem. – Matt Rosenzweig Jun 27 '14 at 17:46
• Yea, I see that, but not sure how to incorporate that into a solution. – user1775614 Jun 27 '14 at 17:48

Let $p$ be the probability of $A$ winning. If $A$ doesn't win with the first move, it is as if $A$ and $B$ had swapped their roles. Thus, $p=\frac{1}{6}+\frac{5}{6}(1-p)$