Well, let's look at this subjectively for a moment:

Let's imagine that you visit your doctor one day, and he tells you that you have cancer. Your doctor is optimistic, and he schedules surgery and chemotherapy to treat your disease. Meanwhile, you are terrified. You don't want to die, so you pray to God day and night for a cure. The surgery is successful, and when your doctor examines you again six months later the cancer is gone. You praise God for answering your prayers. You totally believe with all your heart that God has worked a miracle in your life.
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Now, this is why one would believe in miracles. Now, if we look at it objectively, the results would have been the same if you had prayed to the Flying Spaghetti Monster. See below:

The obvious question to ask is: What cured you? Was it the surgery/chemotherapy, or was it God? Is there any way to know whether God is playing a role or not when we pray?

Unless you take the time to intelligently analyze this situation, it looks ambiguous. God might have miraculously cured your disease, as many Christians believe. But God might also be imaginary, and the chemotherapy drugs and surgery are the things that cured you. Or your body's immune system might have cured the cancer itself.

When your tumor disappeared, in other words, it might simply have been a complete coincidence that you happened to pray. Your prayer may have had zero effect.

How can we determine whether it is God or coincidence that worked the cure? One way is to eliminate the ambiguity. In a non-ambiguous situation, there is no potential for coincidence. Because there is no ambiguity, we can actually know whether God is answering the prayer or not.
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Now, let's look at a situation where we take out the ambiguity:

So what should happen if we pray to God to restore amputated limbs? Clearly, if God is real, limbs should regenerate through prayer. In reality, they do not.

Why not? Because God is imaginary. Notice that there is zero ambiguity in this situation. There is only one way for a limb to regenerate through prayer: God must exist and God must answer prayers. What we find is that whenever we create a unambiguous situation like this and look at the results of prayer, prayer never works. God never "answers prayers" if there is no possibility of coincidence.
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And another:

You can see the same effect in the following prayer. Let's assume that you are a true believer and you do believe that God cures cancer. What would happen if we get down on our knees and pray to God in this way:

Dear God, almighty, all-powerful, all-loving creator of the universe, we pray to you to cure every case of cancer on this planet tonight. We pray in faith, knowing you will bless us as you describe in Matthew 7:7, Matthew 17:20, Matthew 21:21, Mark 11:24, John 14:12-14, Matthew 18:19 and James 5:15-16. In Jesus' name we pray, Amen.
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We pray sincerely, knowing that when God answers this completely heartfelt, unselfish, non-materialistic prayer, it will glorify God and help millions of people in remarkable ways. If God cures cancer, then this is an easy prayer for an omnipotent, all-loving God to answer.

The fact is, what this prayer does is remove ambiguity. As soon as we do that, we see the true nature of "God." There is no way that a coincidence can answer this prayer, and, sure enough, the prayer goes unanswered.
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Now, what does all this mean? God won't answer a prayer if it cannot be ambiguous? Why not? Now, let's look at an article:

In the Duke University study, the researchers studied 748 patients who were undergoing heart procedures such as angioplasty or cardiac catheterization. Congregations of various religions at locations outside the hospital were randomly assigned to pray for half of the patients, without the patients or their doctors knowing which group they fell into.

The patients weren't told because the researchers wanted to separate any impact of prayer from any placebo effect. The prayers followed the traditions of the congregation involved, and continued for five to 30 days. The congregations were told the name, age, and illness of the patient.

Over a six-month period, the study found no difference in serious side effects, death rate, or readmissions between the patients who had received prayers and those who did not.
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Sources: www.godisimaginary.com, http://whywontgodhealamputees.com/ and finally, http://www.boston.com/news/globe/health_science/articles/2005/07/25/a_prayer_for_health/