Let $a, n$ be positive integers with $a > n$. A revolver with $n$ chambers is loaded with $a$ bullets, where the distribution is uniform among all $\binom{n}{a}$ possible choices of $a$ objects filling $n$ slots. 

A lone player is playing [Russian Roulette](https://en.wikipedia.org/wiki/Russian_roulette) using the loaded gun. In this game the gun is spun, which we conceptualise as a chamber being picked uniformly at random. Subsequently, the player fires at his own head without spinning the gun again - which we conceptualise as going through the chambers deterministically in sequence.

In this game, the player intends to shoot himself until he dies.

We denote by $X^{(a, n)}_t$ the survival process of a player using the loaded gun, where at each time $t$, $X^{(a, n)}_t$ takes value $1$ if the player is still alive after the $t$’th round, and $0$ otherwise.

Write $S^{(a, n)} := \sum_{t = 1}^{\infty} X^{(a, n)}_t$ for the survival time of the player.

 
**Question:** Is it true that the mean survival time $\mathbb E[S]$ is *not* monotone decreasing in the ratio $\frac{a}{n}$ of bullets to chambers? 

That is, does there exist $(a, b), (a^{*}, n^{*})$ with $\frac{a^{*}}{n^{*}} > \frac{a}{n}$, but 

$$\mathbb E[S^{(a^*, n^*)}] > \mathbb E[S^{(a, n)}]?$$

Bonus points if it can be shown that $n$ can be taken arbitrarily large.