I was taught that the definite article is used before singular and plural nouns when the noun is specific or particular. First time is a specific noun phrase and therefore requires the article "the" be placed before it. In common usage in North America the following would be correct.
I'm not really expecting to pass the first time.
A student may retake the exam if s/he does not pass the first time.
Don't worry if you can't manage the first time. It'll come with practice.
You should pass the test the first time, but we can't guarantee.