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) $,
>  $$
> \Pr\left[ D^{\left( p\right) }\leq s\,\right] \geq\Pr\left[ D^{\left(
> 1/2\right) }\leq s\,\right] \qquad\forall\, s=1,2,...
> $$
> 
> 
> 



Gambler's ruin: The fair game is the longest
Let us show a bit more, that $F\_p(s):=P(D^{(p)}\le s)$ is nondecreasing in $|p-1/2|$, for any real $s$. That is, take any $p$ and $p\_1$ in $(0,1)$ such that $|p\_1-1/2|>|p-1/2|$, and let, for brevity, $F:=F\_p$ and $G:=F\_{p\_1}$. We shall show that then $F\le G$. 


Indeed, by the [formula (10)](https://www.jstor.org/stable/pdf/3213016.pdf), referenced by Carlo Beenakker, 
\begin{equation\*}
 f\_p(n):=P(D^{(p)}=n)=a\_{n,z} b\_{p,z}(pq)^{n/2} 
\end{equation\*}
for natural $n$, where $a\_{n,z}$ depends only on $n$ and $z$; $b\_{p,z}:=(p/q)^{z/2}+(q/p)^{z/2}$; and $q:=1-p$. Since $pq$ decreases in $|p-1/2|$, the family $(f\_p)$ of pmf's has the [monotone likelihood ratio (MLR) property](https://en.wikipedia.org/wiki/Monotone_likelihood_ratio), meaning that 
\begin{equation\*}
 r(n):=\frac{f(n)}{g(n)}
\end{equation\*}
increases in $n\in N$, where $N:=N\_z:=\{n\colon a\_{n,z}\ne0\}$, $f:=f\_p$, $g:=f\_{p\_1}$, 
and $p$ and $p\_1$ are as before. 
Therefore, for any given real $s$, 
\begin{align\*}
 F(s)-G(s)&=F(s)[1-G(s)]-G(s)[1-F(s)] \\ 
 &=\sum\_{m,n\colon\, N\ni m\le s<n\in N}[f(m)g(n)-g(m)f(n)] \\ 
 &=\sum\_{m,n\colon\, N\ni m\le s<n\in N}[r(m)-r(n)]g(n)g(m)\le0, 
\end{align\*}
since $r$ is increasing. Thus, indeed $F\le G$. 


