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{\bf Andreas Blass and G\'abor Braun}
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{\bf Random Orders and Gambler's Ruin}
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We prove a conjecture of Droste and Kuske about the probability that
$1$ is minimal in a certain random linear ordering of the set of
natural numbers.  We also prove generalizations, in two directions, of
this conjecture: when we use a biased coin in the random process and
when we begin the random process with a specified ordering of a finite
initial segment of the natural numbers.  Our proofs use a connection
between the conjecture and a question about the game of gambler's
ruin.  We exhibit several different approaches (combinatorial,
probabilistic, generating function) to the problem, of course
ultimately producing equivalent results.

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