With high probability

Definition
An event occurs with high probability if, for any $$\alpha \geq 1$$, the event occurs with probability at least $$1 - \frac{c_\alpha}{n^\alpha}$$, where $$c_\alpha$$ depends only on $$\alpha$$.

Since we can choose $$\alpha$$, we can make the probability arbitrarily low, at a cost of time and/or space.

Example
With high probability, a set of N random numbers will contain at least O(N) evens.

What it means is: For any $$\alpha$$, there exists a k (that doesn't depend on N), such that a set of N random numbers will contain at least k*N evens with probability $$1 - \frac{c_\alpha}{n^\alpha}$$, where $$c_\alpha$$ depends only on $$\alpha$$.