With high probability

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Definition[edit]

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[edit]

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