Stephen is right. We should distinguish the Design Argument from the Ontological Argument. Your question concerns neither. Your question is about the Problem of Evil, so called. How can a being who is all-good, all-powerful and all-knowing allow evil to exist? The simplest way to solve this problem is to deny one of these three propositions, and it is perfectly acceptable to deny the second: God's power is limited. This approach is taken by process theologians, who say that God is developing. For the typical process theologian, as for the Mormon, God cannot break the laws of nature, for example. The trouble with this solution is not that it does not work for the theist. The problem is that it does not work for the traditional Christian theist, and as far as I know also for the Jewish and Muslim theist. A god with limited powers is simply not recognizable as the Creator of the Universe, the Father Almighty, and so on. So the solution is logically acceptable, but theologically unacceptable.
Just to add a bit to what my fellow panelists have said (all of which seems right to me.)
Even if God can't be seen or heard or touched in ordinary sensory ways, many believers would claim that they have experiences of God. There's a vast literature on this topic, but one interesting recent contribution is by the anthropologist Tanya Luhrmann: her book When God Talks Back gives a detailed account of how some people, as they understand it, learn to experience God. You can read a brief synopsis HERE (Scroll way down if the page appears not to load properly.)
You might say that these people are mistaken, and you might (or might not) be right. You might say that they are deluded, but unless you simply mean "mistaken," the word "deluded" doesn't add anything. There's no reason to believe that such believers are mentally ill by any reasonable criterion.
As it happens, I'm not a theist. But over the years, I've come to the conclusion that many atheists have a mistaken picture of the religious lives of believers. This leads to a good deal of misunderstanding.
Charles Taliaferro's reply is very helpful. For me the two things that have the most importance for your question are the Wittgensteinian approach, in which the one who wants evidence that the good that happens is, indeed, the result of prayer, is slipping in and then out of the way of the religious attitude to the world. When you have that attitude towards the world, you will not ask the question, or you cannot . . . This is similar to the approach given in John Wisdom's "Gods", in the parable of the gardener. The replies that were made to this piece are equally interesting. I also wonder why one wants to know whether an outcome is indeed the result of prayer. I personally do not have your question, though I am not sure why not, unless it is what I have said above, and so it occurs to me to ask whether the question itself might need a kind of justification, and, if so, what form it would take.
True confessions: like Charles, I accept the Ontological Argument. But it must be said that a response to the Argument based on "the problem of evil" is something of a mistake. The reason is that the problem of evil is a problem for all arguments for theism, and offers nothing specific for us to learn about the ontological argument, particularly its logic, which is where almost all the interesting issues are.
Some random suggestions: (1) David Pears pointed out that even if Kant's argument were wholly clear and wholly successful, which it is not, it could only show that existence is not an ordinary predicate, if it is a predicate. His view is that it is a predicate, just a very peculiar one; (2) There is also the view of the celebrated logician, mathematician and philosopher Bolzano, who writes in the Theory of Science ("Kinds of Propositions") that 'I take being [Sein] or actuality [Wirklichkeit] to be precisely what language makes it out to be, namely an attribute; whoever denies this confuses (I believe) actuality with substance. By substance I mean an actuality which is not an attribute of another actuality; hence I admit that we cannot truly predicate the putative abstractum of the substance (substantiality) of any object. For it is part of the concept of substance that there is no property of this kind. But it is not the same with actuality, which I consider to be a mere attribute, not only of substance itself but of each of its attributes, since every attribute of an actual thing is itself actual. And since every attribute of an object can be ascribed to it in a judgment of the form 'A has b', why not the attribute of actuality?' (3) There is a related argument deriving from Russell's Theory of Descriptions in my own Philosophical Propositions, despite the fact that Russell himself took the implication of the theory to be that the ontological argument is no good; (4) There is a defence of a stripped-down version of the ontological argument by the late Gary Matthews and Lynn Baker Rudder in Analysis for 2010.
This is a profound and difficult philosophical question. I have toyed with the idea that it is wrong to teach children anything normative in the areas of politics and religion - at least they won't know enough to spoil dinner table conversation when they grow up. Seriously, I am not sure what the answer is, but I think that I would want to take my stand on a distinction between teaching by indoctrination and teaching by example. It is difficult to see that there could be an objection to people raising their children in a context in which the faith of the parents is evident. (But what happens when the child is to copy the parent in the recitation of the Nicene Creed? - "We believe in one God, the Father Almighty, Maker of all things visible and invisible . . ."?) And when it comes to explicit religious instruction, things can get sticky. As long as the transmission of the faith is restricted to example and reason, though, I think it is acceptable. When the method of transmission is authority and indoctrination, it is not, as I see it. Something destructive enters the picture.
let me add a bit more in favor of the argument here ... we do tend to believe that certain very improbable things do not occur by chance -- poker/slot machine analogies common -- if your friend gets five royal flushes in a row you'd almost certainly be pulling your piece on him -- the fine tuning argument suggests that the very same sort of very ordinary, accepted reasoning applies to the universe -- that the specific tuning of the various constants is so improbable, when all others are possible (no combination of which would lead to any foreseeable valuable universe, key point), that just as you respond to your poker friend you should respond to the universe: not likely to have occurred by chance (tho always, of coure, remotely possible) -- but still the fact it is remotely possible that your friend randomly drew 5 straight royal flushes would stop no one from reaching for their piece ....
i have a bit more about the argument in my book 'the god question' --
hope that helps!