oone exmple in hash topics?
Let $A_n\subseteq \{0,1\}^{\log n}$ denote the set of binary strings of length $\log n$ with at least $c\log n$ trailing zeros on the right (here I assume that $\log n$ in an integer and $0\le c\le 1$). Note that $|A_n|=2^{\log n -\lceil c\log n\rceil}\le 2^{(1-c)\log n}$. Since $H:\{0,1\}^{\log n}\rightarrow \{0,1\}^{\log n}$ is a one to one function, $\Pr_{x\sim U_{\log n}}\left[H(x)\in A_n\right]=\frac{|A_n|}{2^{\log n}}\le\frac{1}{n^c}$, where $U_k$ is the uniform distribution over $\{0,1\}^k$.