Are there any rules for omitting an article after the word "with"?
for instance, a sentence I see a lot is:
The procedure fails with small probability
rather than with a small probability.
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.
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.
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.
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.
