articles: after "with"

Question

Are there any rules for omitting an article after the word "with"?

1. for instance, a sentence I see a lot is:

   The procedure fails with small probability

   rather than with a small probability.

2. What happens when we have a possession of the form: <object1> with (a/?) <object2> where the first object is plural, but the second is not. E.g.,

   procedures with a fixed cost

   or

   procedures with fixed cost

   ... or maybe a simpler example: cars with radio vs. cars with a radio.

Answer 1

The sentence you posted looks wrong partly because a "probability" is really low-high, not small-large. The correct sentence would be more along the lines of:

The procedure fails with low probability

I know that doesn't answer your question, but without correcting that either answer will "look" a little wrong, making it harder for you to judge the issue of article placement.

Now to the answer: You can miss the article because "probability" isn't really an object. You can't hold a probability, nor can you eat, have, hold or throw one. As such, it's not really something that "needs" an article.

Personally, I'd still use "with a" in that sentence, as I think it's both "more" correct and sounds better.

The procedure fails with a low probability

Then again, I'm still not sure the sentence makes sense :p does probability usually affect an outcome? Or does it simply describe the likelihood of an outcome? That's probably a question for another day.

Answer 2

I'm going to place a bet on the idea that the article differentiates between a defined object and a concept. This doesn't discredit the other answers here which are respectable in their own right.

The concept of 'low probability' is just that: a concept, an idea. To attribute this concept to something like a procedure says that the procedure has a low chance of failing. 'A low probability' implies that there is a defined probability for this object instead of just declaring that there is a low chance of something happening.

For 'procedures with fixed cost' I would include the article, because in this example you would define a fixed cost: a 'fixed cost' isn't really a concept, nor is it the same for each procedure.

The same follows with the car example: each car has its own radio, hence 'cars with a radio', or, even better, 'cars with radios'.

TL;DR:

The inclusion of the article implies a defined object instead of the concept of one. Consider the difference in meaning between 'drugs' and 'some drugs', or 'alcohol' and 'some alcohol' or 'friendship' and 'a friendship'.

Trivia: It's the opposite in Spanish. 'Las drogas' is the concept while 'drogas' is the objects.

Answer 3

Hmmm there are interesting answers here. However, I would try to forget the differentiation between concepts (like probability) and physical constructs (like a probability). These are not that well defined in English grammar for anyone to prescribe the correct usage of. That being said, as a descriptive grammar I would agree with @Jon Story.

The prevalence is to use the article - stick with that and you will not go far wrong.