How Does Prayer Work? Part 1

How exactly does prayer work? I know that there are different kinds of prayer: the talking-to-God prayer, the Praise-and/or-thanks-to-God prayer and others, but I'm mainly talking about the I-want-something prayer where you ask your god for something and when you get it you say "prayer works!"

Now I'm not saying that one shouldn't pray or that prayers aren't answered, but am asking how one is to know that prayer is indeed working.

In a typical prayer, a person asks God to please provide him with 'xyz'. Sometimes the thing asked for is received, or occurs. This is presented as proof, or at least an indication, that "prayer works". Now one might postulate that if 'xyz' doesn't happen, then prayer doesn't work. But this is seldom that case. There are a variety of responses to a non-appearance of prayer requests.

1. Ignore the non-response. Only draw attention to desired outcomes, so if you get 1 answer out of 100 tries, the success rate isn't 1%, it's 100%, because the one positive result is the only one you're paying attention to.
   
2. Make excuses for your deity. Say that he answers in his own time, or that he works in mysterious ways.
   
3. Father (God) knows best. Point to something else that happened that could be construed as better than what was prayed for. God obviously knew that you needed abc more than xyz, so that's what he gave you.
4. Lowered expectations. Accept part of what you asked for, or a too-late receipt of your prayer request and credit God for answering your prayer.
   

So, with all of that in mind, just how does this prayer thing work?