Probability of first actor winning a "first to roll seven with two dice" contest?

Two players P and Q take turns, in which they each roll two fair and independent dice. P rolls the dice first.

The first player who gets a sum of seven wins the game. What is the probability that player
P wins the game?

Answer

I'm going to assume P goes first. The question should have said so, unless something else was intended, in which case it's unclear.

Let (lower-case) $p$ be the probability that P ultimately wins. Then

$$\begin{align} p & = \Pr(\text{P wins on 1st trial}) + \Pr(\text{P loses on first trial and ultimately wins)} \\[8pt] & = \frac 1 6 + \frac 5 6 (1-p). \end{align}$$
So solve the following:
$$p =\frac 1 6 + \frac 5 6 (1-p).$$