I like that Joe....OUr prayers ought to be humble...So what is humble? Humility is simply bowing to Gods will...Not a fake thingy like "I'm so unworthy." Humility is not contradicting Gods Word. And no, you don't have to quote chapter and verse at Him but your requests should line up at least with His Word. Wouldn't you say?
I'd say it depends on the situation,for example many people around the world say simple prayers even without "perceiving them" like "please God strengthen me" when a person feels weak,or perhaps the most common one "God help me"
when a person is about to be killed in some manner hoping for God's intervention whether they truly believe in God or not is of no consequence,God is no respectee to persons and will help whom he sees fit to help,through answering a prayer with some form of action,quite honestly not everyone has a "bible" or some don't know how to read,so while I get your sentiment and respect it,prayer to God is not turned away by God so easily,think of the king Saul,at one point God no longer answered his prayers,because soon David would be the new king of Israel and Saul continually did wrong by God,so prayer to God by "Anyone" whether biblically inspired or not is acceptable to God if they at that moment in time they are "sincere" and "humble" in their prayer,meaning in their "heart" they "needed" God's help and so therefore God from there decides whether to intervene or not,because God knows what you "need",and what you can "manage" many are disappointed when God doesn't give them what they "want" and they misunderstand it's not that God doesn't already give you "plenty" it's more of we can't decide what we really "need",many think on fame,fortune or the like but those aren't "needs" they are "wants",like I think on Solomon he asked God for "wisdom" and the "knowledge" to judge between people in matters,and because of choosing "wisdom" over money,fame,power,and many other things,God gave him all the things he did not ask for along with what he asked for,prayer is more of a way to tell "honestly" ask God for help concerning one's self or others,or perhaps more plainly put,a way to let God know you need him at the same time as asking for his help,like daniel he prayed 3 times a day,not necessarily for help but more in showing God he cared for him and appreciated him,alot of people forget that prayer is not just for "asking for help" it's also for "speaking to God in appreciation" a prayer isn't always to be about "heal that person" or "help me" it's also about bringing yourself before God and "honestly" telling him how much you appreciate all you have as well as all God does for you,what I'm getting at is that the "Bible" is "important" but prayer is "always" accepted by God so long as you are "sincere" and "humble" and does not have to be silently to God in can be outloud as well as I showed by examples,God listens "always" and there is "almost" no reason that God will ever not answer a prayer such a thing is "nearly" unheard of(Saul is an example but it's "rare" for God to not answer)it wouldn't matter if you were a person that had the least knowledge of the entire world,God always "listens" and "answers" prayers,either yes or no,if God doesn't answer in anyway then the individual would have to be persistently "rebellious" in someway toward God to have such a "rare" result!(as with king saul)