A bug crawls along the edges of a regular tetrahedron $ABCD$ with edges length $1$.

It starts at $A$ and at each vertex chooses its next edge at random (so it has a $1/3$ chance of going back along the edge it came on, and a $2/3$ chance of going along each of the other two).

Find the probability $P$ that after it has crawled a distance of $7$ it's again at $A$.


Let $p_n$ denote the probability that the bug is at $A$ after crawling a distance $n$. Then the recursive formula is $$p_{n+1}=\frac{1}{3}(1-p_n)$$

with intial condition $p_0=1$.

The $(1-p_n)$ is the probability that the bug is not at $A$ after a distance $n$ (and thus has a chance to return to $A$ the next step), and the factor $\frac{1}{3}$ is the probability that the bug goes from any other vertex to $A$.

Solve this recursion or iterate to find $p_7$.

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