Let $X$ be a random variable with mean $\mu$, where $0 < \mu < 1$.
Let $X(n)$ be the sum of $n$ independent ,identically distributed, $X$ variables.
Under what conditions on $X$ , possibly dependent on $n$, is the following result true:
$$P[ X(n) \ge (n+1)] < P[ X(n-1)\ge n]$$
For a simple illustration , if $X$ is discrete taking the values $0$ and $2$ with probabilities $0.7$ and $0.3$ respectively and $n=2$, the result is true as $0.09 < 0.3$.
The general question arose from considering strategies for the players in the weaker of two chess teams in a match.