This is a two part argument. The first argument is for the existence of God based upon the problem of evil. The second, related argument is for the moral obligation to believe in God.
The first argument:
1. The problem of evil exists only if God exists. Phrased negatively, if God doesn't exist then the universe exists exactly as it must and cannot be described with moral terms such as either "good" or "evil." As the atheist Richard Dawkins has said,
In a universe of blind physical forces and genetic replication, some people are going to get hurt, other people are going to get lucky, and you won't find any rhyme or reason in it, nor any justice. The universe we observe has precisely the properties we should expect if there is, at bottom, no design, no purpose, no evil and no good, nothing but blind, pitiless indifference.2. Evil is a problem. We judge that the world ought and should be otherwise.3. Therefore, God exists.
The second argument:
The provocation of these arguments is to make the point that the problem of evil is often taken to be the best argument for atheism. Logically, though, as I try to illustrate above, the problem of evil presupposes theism. We can only make the judgment--either rationally or emotionally--that the world ought to be otherwise if it indeed can be otherwise. This is precisely what Christians believe in the face of evil, that the world ought to be otherwise because it can and will be otherwise. This is a call to moral exertion in the face of evil. By contrast, the position of atheistic materialism is that the world can't be otherwise. The world exists exactly as it must exist, and no configuration of it can be morally judged as either good or evil. The problem of evil simply doesn't exist.1. We are morally obligated to believe that evil is a problem. To look upon evil and say that it is neither good or evil is sociopathic and wicked. We must believe that evil is a problem and name it accordingly.2. Being morally obligated to believe that the problem of evil exists, we are, therefore, morally obligated to believe that God exists.
But what about the problem of pain and horrific suffering? Well, again, from the position of materialism consciousness is epiphenomenological, a mere by-product of particular material arrangements with no causal power upon those material constituents. That some material configurations are associated with conscious pain and torment is the unproblematic way the universe just happens to exist, and can't really be otherwise. Horrific pain is as morally unproblematic as a rock or the law of gravity. As Dawkins points out, according to materialism suffering is "precisely" what "we should expect." So, resign yourself. Can't be otherwise. Move along, there is no problem here.
Of course, though, we can't move along. No caring person could. We judge that a universal moral obligation is involved here. Indifference isn't an option. Which moves us decisively from an atheistic to theistic framework and the moral obligation to believe in God.
Stated simply, because Christians judge that the world ought and can be otherwise, yes, we have a problem of evil. But this is much better, morally speaking, than thinking evil is not a problem, which is the metaphysical implication of materialistic atheism.