Random Orders and Gambler's Ruin

  • Andreas Blass
  • Gábor Braun


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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