The answer is that the Bible suggests mixed things about prayer. In John 14:13 Jesus tells his disciples that he will do whatever they ask in his name. If we take the approach that what Jesus tells his disciples he also tells us then our prayers should also be answered in this way.
We can compare this with Paul's thorn in the flesh in 2 Corinthians 12 where we read that Paul pleaded with the Lord three times to take his thorn in the flesh away from him, but Jesus doesn't and replies that his grace is sufficient for Paul. This seems a different outcome to the one Jesus told his disciples in John.
If we take an alternative approach which is that we need to put limits on the verse in John, i.e. that John records a specific instance when Jesus was speaking about a special period or to a special group then why does anything Jesus say apply to us since he doesn't address us directly, and how do we know what is limited and what isn't? If it's some half-way house where some of what Jesus says applies now, but other parts don't then how do we discern which is which from the Bible? I don't think we can.
I have heard Christadelphians assert that our prayers are answered in one of three ways; 'Yes', 'No' or 'Not yet'. I think this is an interesting manoeuvre for two reasons. Firstly, because it means that there is no evidence that can possibly be used to falsify this assertion - as a 'truth proposition' it is impossible to verify. And secondly, because if prayer didn't work we would expect to see these exact same outcomes. If the natural course of events were to play out then these are the only three possible options available.
I want to include some thoughts on the efficacy of prayer. Many studies have been done to see if prayer works, there is even a wikipedia page that catalogues some of this research. The research that has been done, especially the most statistically rigorous studies, show there is no differences between those who are prayed for and those who are not. This seems to falsify the position that prayer is effective in changing physical world outcomes.
What about if prayer is not about our own desires and God's will overrules? If this is the case then why pray anyway? There seems little point praying for something if it isn't going to change an outcome. This is also not how the Bible describes prayer working in, for example, Isaiah 38 when Isaiah comes to Hezekiah and tells him that he is going to die. Isaiah tells Hezekiah, "This is what the LORD says...you are about to die", Isaiah then leaves and Hezekiah prays to God. Immediately Isaiah is told to go back to Hezekiah and tell him, "This is what the Lord God says: “I have heard your prayer...I will add fifteen years to your life,"
My personal experience of prayer is that it doesn't change physical world outcomes and that there is no reason to believe that the prevailing natural course of events has been changed.