Let's say I have a real valued random variable $X$ distributed according to a continuous non normal cdf $G(x)$ with mean $\mu$, standard deviation $\sigma$ and pdf $g(x)$. Furthermore I have a set of natural number $[k]$ where $\{k \in \mathbb{N} : 0 < k < 30\}$. Now I go on as follows:
From $[k]$ I randomly sample one number $K$
From $X$, I sample my first random sequence of size $K$: $S_1=\{x_{1,1}, x_{1,2}, ..., x_{1,K}\}$ with sample mean $\bar{S_1}=\frac{1}{K}\sum_{k=1}^K x_{1,k}$.
From $X$, I sample my second random sequence of size $K$: $S_2=\{x_{2,1}, x_{2,2}, ..., x_{2,K}\}$ with sample mean $\bar{S_2}=\frac{1}{K}\sum_{k=1}^K x_{2,k}$
What is now the probability that $\bar{S_2} < \bar{S_1}$?
Take my answer with a grain of salt, but wouldn't it be $ \frac{1}{2} $? I assume that the sampling is an independent event in your case. In that case, since $ \overline{S}_1 $ and $ \overline{S}_2 $ are i.i.d., the chance that $ P(\overline{S}_2 < \overline{S}_1) = P(\overline{S}_2 > \overline{S}_1) = P(\overline{S}_2 \geq \overline{S}_2) $, where the last equation holds since $ \overline{S}_1, \overline{S}_2 $ are continuously distributed (at least I assume you meant that with "real valued random variable X"). Since $ 1 = P(\overline{S}_2 < \overline{S}_1) + P(\overline{S}_2 \geq \overline{S}_2) $, it follows in total that $ P(\overline{S}_2 < \overline{S}_1) = P(\overline{S}_2 \geq \overline{S}_2) = P(\overline{S}_2 > \overline{S}_2) = 0.5 $