Is square of probability $p^2$ is less than probability $p$?

Question

suppose we have $N$ possible states in system. There is a probability $p_n$ that system is in state $|n\rangle$, and the sum of all probabilities is one. Is there any general rule in math or physics for which the squared probability $p_n^2$ is less than probability $p_n$?

Answer 1

The sum $\sum_n p_n^2 \le \sum_n p_n$ with the equality only being if $p_n = \delta_{nk}$ for some $k$. This follows because $p_n \in [0, 1]$ (and hence $p_n^2 \le p_n$) and $\sum_n p_n = 1$.