# "With probability", "with a probability", "with the probability"?

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?

• 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%. . Apr 20, 2021 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. Apr 20, 2021 at 10:59
• 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. Apr 20, 2021 at 12:52
• 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). Apr 20, 2021 at 15:27