For Which General Distributions Does This Inequality Hold? Let $X$ be a  random variable with mean $\mu$, where $0 &lt; \mu &lt; 1$.

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:

$$P[ X(n) \ge  (n+1)]  &lt;  P[ X(n-1)\ge n]$$

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 $0.09 &lt; 0.3$.

The general question  arose from considering strategies for the players in the weaker of two  chess teams in a match.
X having a Normal distribution with mu &lt; (3-2^0.5)/(2-2^0.5) , implicitly n>=2 will do: consider z scores.

Alternatively X (negative)  exponential with Erlangian sums will do for appropriate mu. 

Other examples may well be possible
X having a Normal distribution with mu &lt; (3-2^0.5)/(2-2^0.5) , implicitly n>=2 will do: consider z scores.

Alternatively X (negative)  exponential with Erlangian sums will do for appropriate mu. 

Other examples may well be possible