Duration of gambler's ruin conditioned on a chosen boundary (${\mathbb{E}}_i[{\tau }_0\mid {\tau }_0<{\tau }_r]=\frac{i(2r-i)}{3}$)

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 $g_i={\mathbb{E}}_i[\tau 1_{\{{\tau }_0<{\tau }_r\}}]$, 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.