# With high probability

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## Definition[edit]

An event occurs **with high probability** if, for any , the event occurs with probability at least , where depends only on .

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

## Example[edit]

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

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