  In this paper we provide formulas for the expectation of a conditional game
duration in a finite state-space one-dimensional gambler's ruin problem with
arbitrary winning $p(n)$ and losing $q(n)$ probabilities (i.e., they depend on
the current fortune). The formulas are stated in terms of the parameters of the
system. Beyer and Waterman [Mathematics Magazine, 50(1), 1977] showed that for
the classical gambler's ruin problem the distribution of a conditional
absorption time is symmetric in $p$ and $q$. Our formulas imply that for
non-constant winning/losing probabilities the expectation of a conditional game
duration is symmetric in these probabilities (i.e., it is the same if we
exchange $p(n)$ with $q(n)$) as long as a ratio $q(n)/p(n)$ is constant. Most
of the formulas are applied to a non-symmetric random walk on a circle/polygon.
Moreover, for a symmetric random walk on a circle we construct an optimal
strong stationary dual chain -- which turns out to be an absorbing,
non-symmetric, birth and death chain. We apply our results and provide a
formula for its expected absorption time, which is a fastest strong stationary
time for the aforementioned symmetric random walk on a circle. This way we
improve upon a result of Diaconis and Fill [The Annals of Probability, 18(4),
1990], where strong stationary time -- however not the fastest -- was
constructed. Expectations of the fastest strong stationary time and the one
constructed by Diaconis and Fill differ by 3/4, independently of a circle's
size.
