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As a mathematician, I often write and read about probabilities. In the literature, I've seen versions of all the sentences below. Which one is correct?

This happens with probability (of) 30%.

This happens with the probability (of) 30%.

This happens with a probability (of) 30%.

In particular, what article should one use and is "of" correct/necessary?

Paula
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    When I hear a weather report, it's of the form The chance of rain is 30%. When I read a math / statistics paper, I have seen With probability 0.30 and With a probability p=0.30 or With a p-value = 0.0235. I have rarely seen the definite article, e.g., With the probability of 0.30 simply because mathematicians rarely talk about 30% as the "one-and-only" 30%. . – rajah9 Apr 20 '21 at 10:56
  • English usage is at our question about articles. This appears to be jargon, though: a field-specific usage for mathematical academic literature. – Andrew Leach Apr 20 '21 at 10:59
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    Here's an NGram chart querying all relevant permutations. Showing that with a probability of 25% beats out article-less with probability 25% - the others (with a probability 25% with the probability 25%, with the probability of 25%) are all just too rare to chart. – FumbleFingers Apr 20 '21 at 12:52
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    Agree with the comments above. Would add that when you give the shorter version ("The event occurs with probability 30%") you get a marginally more scientific/technical sound than when you use "a... of" (to my ears at least). – cruthers Apr 20 '21 at 15:27
  • with a probability of 30% OR with 30% probability. – Lambie Jun 01 '23 at 15:40

2 Answers2

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Dear fellow mathematician!

Here is a math question: Output and interpret the prediction interval for the one-step forecast of your chosen model.

I answered:
hilo(fc) %>% select(Date, '80%')
With probability 80%, the one-step future value will fall in that prediction interval.

I got points deducted and the feedback from Prof. Edwards:
'''
You can't say "with probability" when interpreting a confidence interval or prediction interval.  You can only say that if you are doing a Bayesian analysis.
'''

I had learnt game theory as an economic student where the phrase "w.p." had been used all the time and was not most familiar with that rigorous math terminology when I recently made a mistake in a math course.

Dorian Merci
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    As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Jun 01 '23 at 12:48
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    This is a different issue, Dorian: 'Is the term "probability" correct in this context?' OP asks about which article to use, and whether to include 'of'. Yours is a maths/stats or even philosophy question. Of course, 'probability' needs to be rigorously defined locally before one can even start an analysis. Better asked on Mathematics.SE. – Edwin Ashworth Jun 01 '23 at 13:17
  • As people have answered "With probability 0.30" and "With a probability p=0.30" are acceptable -use no article or indefinite article, I just want to in addition emphasis the correct use of the phrase "with probability". If the context of "probability" is frequentist instead, expressions like 30% of the times should be correct. – Dorian Merci Jun 02 '23 at 08:12
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I would say that:

This happens with probability 30%

Here adding 'of' would be ungrammatical. Whilst

This happens with a probability of 30%

Here the 'of' is necessary for the sentence to be gramatically correct.

Mozibur Ullah
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    Hello, Mozibur. While I agree that these sound the most natural (and my degrees are in science, and I taught maths/statistics), ELU expects answers to be backed by linked and attributed supporting references. // I'm not sure that this warrants the downvote, though. It would often be better if downvotes required reasons to be added. – Edwin Ashworth Apr 20 '21 at 09:32