4-partition elements summation NP completeness
Let's say that the numbers $a_1,\ldots,a_n$ are non-negative, and sum to $A$. They induce a probability distribution on $\{1,\ldots,n\}$, in which the probability of $i$ is $a_i/A$.

Given a partition $S_1,S_2,S_3,S_4$, let $I$ be a random variable distributed as above, and let $X$ be the index of the set to which $I$ belongs (so $X \in \{1,2,3,4\}$).
Your condition then reads $$ H(X) \geq K, $$
where $H(\cdot)$ is the entropy function.

This suggests choosing $K = 2$, since we know that $H(X) = 2$ iff $\Pr[S_1] = \Pr[S_2] = \Pr[S_3] = \Pr[S_4]$.

We can now describe a reduction from PARTITION to your problem. Given an instance $b_1,\ldots,b_m$ of PARTITION, where $b_1,\ldots,b_m$ are non-negative integers summing to $B$, create an instance of your problem by adding two copies of $B/2$, and by taking $K=2$. The list $b_1,\ldots,b_m,B/2,B/2$ can be partitioned into four parts with equal sum (i.e., $H(X) \geq 2$) if $b_1,\ldots,b_m$ can be partitioned into two parts with equal sum.