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In their paper "Handling Churn in a DHT", the authors use the following sentence:

For each event we select a node to die uniformly at random[...]

Do you think this sentence is unprecise? Or is it even grammatically wrong? I argued with a friend wether someone could misinterpret this sentence (e.g. that nodes are dying uniformly) or if the sentence is as precise as needed.

An alternate form could be

For each event we randomly uniformly select a node to die [...]

We believe the syntax of this sentence to be correct and that it is more precise than the first version, but IMHO it lacks readability.

Is one of the forms above to be preferred over the other, or would you say that both aren't optimal and that there are better ways?

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    In this case, "uniformly" appears to apply to the "die" process. "Random" appears to apply to the "select" process. Apr 22 '13 at 14:16
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The phrase "uniformly at random" is a very common phrase in probability theory, and people in the field will understand it, even if it isn't precise if you read it as an ordinary English sentence.

If you want to replace it, you should use "randomly and uniformly", which is also a very common phrase in probability theory. Why do you need an "and"? These two adjectives have the same place in natural adjective order, so they are coordinate adjectives and not cumulative adjectives; this means they want an "and" to separate them. (And furthermore, the mathematical community usually uses an "and" here.) See this question about using "and" between adjectives.

If you're worried that your readers might not understand "uniformly at random", there is nothing wrong with replacing it with "randomly and uniformly".

Here is a Google Ngram in support of these statements.

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