Consider a point P moving on the coordinate plane every 1 second according to the following rules (i), (ii).
[(i)] Initially, P is at the point $(2,\ 1)$.
[(ii)] When P is at point $(a,\ b)$ at some moment, 1 second later P is at
the point symmetric to $(a,\ b)$ with respect to the $x$-axis, with probability $\dfrac{1}{3}$
the point symmetric to $(a,\ b)$ with respect to the $y$-axis, with probability $\dfrac{1}{3}$
the point symmetric to $(a,\ b)$ with respect to the line $y = x$, with probability $\dfrac{1}{6}$
the point symmetric to $(a,\ b)$ with respect to the line $y = -x$, with probability $\dfrac{1}{6}$
Answer the following questions. For question (1), it suffices to state only the conclusion.
[(1)] Find all possible coordinates of points that P can occupy.
[(2)] Let $n$ be a positive integer. Show that the probability that P is at point $(2,\ 1)$ after $n$ seconds from the start equals the probability that P is at point $(-2,\ -1)$ after $n$ seconds from the start.
[(3)] Let $n$ be a positive integer. Find the probability that P is at point $(2,\ 1)$ after $n$ seconds from the start.
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\noindent\textbf{3}
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Consider a point P moving on the coordinate plane every 1 second according to the following rules (i), (ii).
\begin{itemize}
\item[(i)] Initially, P is at the point $(2,\ 1)$.
\item[(ii)] When P is at point $(a,\ b)$ at some moment, 1 second later P is at
\begin{itemize}
\item the point symmetric to $(a,\ b)$ with respect to the $x$-axis, with probability $\dfrac{1}{3}$
\item the point symmetric to $(a,\ b)$ with respect to the $y$-axis, with probability $\dfrac{1}{3}$
\item the point symmetric to $(a,\ b)$ with respect to the line $y = x$, with probability $\dfrac{1}{6}$
\item the point symmetric to $(a,\ b)$ with respect to the line $y = -x$, with probability $\dfrac{1}{6}$
\end{itemize}
\end{itemize}
\noindent Answer the following questions. For question (1), it suffices to state only the conclusion.
\begin{itemize}
\item[(1)] Find all possible coordinates of points that P can occupy.
\item[(2)] Let $n$ be a positive integer. Show that the probability that P is at point $(2,\ 1)$ after $n$ seconds from the start equals the probability that P is at point $(-2,\ -1)$ after $n$ seconds from the start.
\item[(3)] Let $n$ be a positive integer. Find the probability that P is at point $(2,\ 1)$ after $n$ seconds from the start.
\end{itemize}
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