# 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 cα depends only on α.

Since we can choose α, 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 Ω(N) evens.

What it means is: For any α, 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 cα depends only on α.