= Solution
Let $h_k$ be the <hitting probability> of the right endpoint before the left endpoint, starting at $k$. A surviving vase corresponds to that right-endpoint event. Conditioning on the first step of the <simple symmetric random walk> gives
$$
\boxed{h_k=\frac12h_{k-1}+\frac12h_{k+1},\quad 1\le k\le19;\qquad h_0=0,\quad h_{20}=1.}
$$
The <Markov property> makes the remaining probability depend only on the position after that step. These absorbing endpoint values distinguish the event of interest from its complement.
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