I was just giving a cruddy distillation of what I was taught by my UCB professor haha. If he's wrong or I am wrong in understanding the example he gave from the textbook, here is what was given to me in the textbook (I hope this isn't a distraction; thanks for the response):
Suppose that Smith and Jones have applied for a certain job. And suppose that Smith has strong evidence for the following conjunctive proposition: (d) Jones is the man who will get the job, and Jones has ten coins in his pocket. Smith's evidence for (d) might be that the president of the company assured him that Jones would in the end be selected, and that he, Smith, had counted the coins in Jones's pocket ten minutes ago. Proposition (d) entails: (e) The man who will get the job has ten coins in his pocket. Let us suppose that Smith sees the entailment from (d) to (e) and accepts (e) on the grounds of (d), for which he has strong evidence. In this case Smith is clearly justified in believing that (e) is true. But imagine, further, that unknown to Smith, he himself, not Jones, will get the job. And also, unknown to Smith, he himself has ten coins in his pocket. Proposition (e) is then true, though proposition (d), from which Smith inferred (e), is false. Gettier concludes that in this case Smith has justified true belief in (e) but doesn't know (e) to be true. It's a matter of luck that he is correct. Other terms like "accidentally correct" or "correct as a matter of sheer coincidence" apply as well. (McGrath, Epistemology: A Contemporary Introduction)
I think your synopsis represents one of Gettier’s two original(?) problems well. The red barn Gettier problem (Goldman) is a more serious consideration. Other cases are also more serious. As a common denominator, fallibility and luck are in play. But philosophers have had the good sense to eliminate arbitrariness from the evidences. For instance the lucky justification in “red barn” is based upon *strong* evidence, unlike the arbitrary and irrelevant correlation of ten coins that is linked for no apparent or obvious reason to the truly reasonable justification for thinking Jones will get the job based upon the boss’ say so. Conjunctive propositions muddy the water and miss the force of the problem in my opinion.