= Absorbing four-ball Ehrenfest chain
{title2=$h=(1,5/8,1/2,3/8,0),\quad t=(0,7,8,7,0)$}
Stop the four-ball <Ehrenfest urn model> upon reaching count zero or four. From count $r$, the transitions are to $r-1$ with probability $r/4$ and to $r+1$ with probability $(4-r)/4$. <First-step analysis> with absorbing boundary values gives the displayed vector $h$ of probabilities of absorption at zero and vector $t$ of <expected hitting times> of either endpoint. For instance, $t_1=1+3t_2/4$ and $t_2=1+(t_1+t_3)/2$, with $t_1=t_3$, give $t_1=7$. From any interior state, absorption has probability at least $1/4$ within two steps; the resulting geometric tail bound proves both almost-sure absorption and finiteness of the expectations.
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