Consider a gambler who, in every trial of a game, wins or loses a dollar with probability $ p\in\left( 0,1\right) $ and $ q=1-p$ , respectively. Let his initial capital be $ z>0$ and let him play against an adversary with the same capital $ z>0$ . The game continues until one of the players is ruined.

In the language of random variables, this amounts to consider a sequence $ X_{1}^{\left( p\right) },X_{2}^{\left( p\right) },…\ $ of random variables (on a sample space $ \Omega$ ) taking on two values $ +1$ and $ -1$ with probabilities $ \Pr\left[ X_{n}^{\left( p\right) }=+1\right] =p$ and $ \Pr\left[ X_{n}^{\left( p\right) }=-1\right] =q$ . In particular, $ X_{n}^{\left( p\right) }$ describes the gambler’s gain on the $ n$ th trial and $ $ S_{n}^{\left( p\right) }\left( \omega\right) =X_{1}^{\left( p\right) }\left( \omega\right) +\cdots+X_{n}^{\left( p\right) }\left( \omega\right) ,\quad S_{0}^{\left( p\right) }\left( \omega\right) \equiv0\qquad\forall \,\omega\in\Omega $ $ describes his net cumulated gain after $ n$ trials. The duration of the game is the random number of trials before he is either ruined or wins the game (and his adversary wins or is ruined) $ $ D^{\left( p\right) }\left( \omega\right) =\min\left\{ t\in\mathbb{N}% :\left\vert S_{t}^{\left( p\right) }\left( \omega\right) \right\vert =z\right\} \qquad\forall \,\omega\in\Omega. $ $

Q. I need a reference or a simple proof of the folk result according to which, for each $ p\in\left( 0,1\right) $ , $ $ \Pr\left[ D^{\left( p\right) }\leq s\,\right] \geq\Pr\left[ D^{\left( 1/2\right) }\leq s\,\right] \qquad\forall\, s=1,2,… $ $