6. Person A and Person B are playing a game of taking turns to toss a coin. The winner of each round is the first one to toss a head (one side of the coin has a head pattern, the other side has a landscape pattern). They agree that the winner of this round will toss second in the next round. Suppose A tosses first in the first round. Then the probability that A wins the seventh round is $\qquad$

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
6. $\frac{1094}{2187}$.

For each game, the probability of the first player winning is
$$
\frac{1}{2}+\left(\frac{1}{2}\right)^{3}+\left(\frac{1}{2}\right)^{5}+\cdots=\frac{\frac{1}{2}}{1-\frac{1}{4}}=\frac{2}{3} \text {, }
$$

and the probability of the second player winning is $1-\frac{2}{3}=\frac{1}{3}$.
Let $P_{k}$ denote the probability that player A wins the $k$-th game, with the recurrence relation
$$
\begin{array}{l}
P_{k}=\frac{1}{3} P_{k-1}+\frac{2}{3}\left(1-P_{k-1}\right)=\frac{2}{3}-\frac{1}{3} P_{k-1}, \\
P_{1}=\frac{2}{3},
\end{array}
$$

which simplifies to $P_{k}=\frac{1}{2}+\frac{(-1)^{k-1}}{2 \times 3^{k}}$.
Thus, $P_{7}=\frac{1094}{2187}$.