= Solution
After $j$ choices, let $C_j$ be the number of forward moves. The two unwrapped positions are $C_j$ and $F-(j-C_j)$, whose gap is $F-j$. It follows that the first subsequent meeting occurs exactly at step $F$, regardless of the order of the choices. Its <position> is the number of forward moves in that block.
That count has a <binomial distribution> with parameters $F,p$. Therefore
$$
\boxed{p_n^{\rm meet}=\binom Fn p^nq^{F-n},\qquad 0\leq n\leq F.}
$$
The two endpoint labels $0$ and $F$ represent the same physical seat reached by different complete circuits; the unwrapped convention distinguishes their <probabilities>.
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