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Prayer is communicating with God and fellowship with him. Just like we talk to our children, family and friends daily God wants that same communication, fellowship, attention and love. And remember He's not only our God, but He is also our heavenly Father.
Whether God causes or allows, it makes no difference. He is in control of everything. When I pray to God, I only ask for understanding, wisdom, and strength. He has never failed me yet.
First and foremost, we pray because God tells us to pray. To paraphrase an old saying, "When God says 'Jump!', you don't ask 'Why?', you ask, 'How high?'" There are many reasons given in scripture for us to pray, but the main reason is because God wants us to pray.
When we pray within God's will, we have relationship with Him. We are agreeing with Him that what HE wants is also what WE want. He has His own plans and purposes, and will do as He has chosen to do. But He wants us to pray so that we have communication with Him. We build relationship with Him.
Imagine a parent with a young child. The parent makes a decision to bake some cookies. The cookies will be made one way or another because that is what the parent intends to do. But the parent loves their child, and they want to spend time with the child, sharing with them the experience, creating loving memories and building and strengthening their mutual relationship. The parent can make the cookies efficiently, quickly, and with delicious result. The parent knows that including the child with mean it will take longer, be messier, with too much butter, and maybe a little egg shell dropped in the mix. But what joy they will share letting those little hands crack the eggs, spill the flour, and getting to lick the beaters!
God loves us and wants us to be a part of all that He does. He doesn't require our input, but He was to share with us so we feel the joy of that relationship. THAT is why we pray!
I don't pray for any of the reasons you mentioned. I don't pray because God said for us to pray. He told us, "Love our enemies." I don't. He told us to do lots of things that I haven't figured out how to do. I pray for a better understanding of how to do those seemingly impossible things. He said to the disciples, "Without me you can do nothing," (Jn 15:5) so I think that goes for me too.
I also pray to stay in touch. He lives in me but He keeps a low profile, so I have to get Him stirring in me through prayer.
Then there are the people I want Him to watch over. My relatives, friends, my community, the homeless, the blind, lame and disenfranchised; the bereaved, the sick, and the shut-ins all need God's attention.
The question is: Do any of those prayers change anything? Do they do any real good? I don't know about anybody else, but it gives me comfort to talk to the One who knows the very hairs of our heads (Lk 12:7). Why would He keep tabs on us like that if He didn't really care a lot about our condition?
Then why doesn't He fix everything that's all messed up? Why war, abortion, sickness, disease, homelessness, and the list is almost endless. If He answers prayer why does all the above continue when we collectively, constantly beg Him to do something about it?
Wars end. Sickness ends. Diseases are permanently wiped out. Homeless people find housing. Women change their minds about an abortion. The answers to prayer are endless.
God works on a case by case basis.
So we pray...