= Duration of gambler's ruin conditioned on a chosen boundary
{title2=$\mathbb E_i[\tau_0\mid\tau_0<\tau_r]=\frac{i(2r-i)}3$}
For a simple symmetric random walk absorbed at $0,r$, the probability of absorption at $0$ is $h_i=(r-i)/r$. Conditioning on that absorption gives the <Doob h-transform> with transition probabilities $(r-i+1)/[2(r-i)]$ downwards and $(r-i-1)/[2(r-i)]$ upwards. If $g_i=\mathbb E_i[\tau\mathbf1_{\{\tau_0<\tau_r\}}]$, first-step analysis gives $g_i=h_i+(g_{i-1}+g_{i+1})/2$, $g_0=g_r=0$. Its solution is $g_i=i(r-i)(2r-i)/(3r)$, and dividing by $h_i$ gives the conditional duration.
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