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I am not sure how to read little/big O expressions in probability theory:

What does a statement like "with probability $1-o(1)$" mean? Does it mean with high probability?

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    http://en.wikipedia.org/wiki/Big_O_in_probability_notation2012-04-30

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It means that the probability tends to 1 if something goes to a limit (usually the size of the sample/object goes to infinity). Without some more context nothing more can be said.