There is no logical problem of evil. 1 and 2 are not logically contradictory. See here for the post on this argument.
An all-powerful and all-loving God exists.
Evil exists.
The proponent of the problem of evil (POE) however does have another option. They might say that though the idea of an infinitely powerful and loving God is not contradictory to the existence of evil in the world, it is nevertheless improbable that both 1 and 2 are true. In other words, they can seek to show that though God’s existence is not impossible, it is nevertheless extremely unlikely given the amount of evil and suffering in the world.
Three considerations are available that can offset the force of this argument.
Probabilities must always be assessed with respect to the background knowledge. If evil were to be taken in isolation, then the theist could freely admit that it provides grounds for thinking God’s existence is improbable. However, the theist should insist that evil be assessed relative to the full scope of evidence for God’s existence.
We are not in any position to know or assess if God has no morally sufficient reason for permitting the evils in the world. God’s foreknowledge extends perfectly into the distant future, while we are limited creatures who can only guess at the ripples effects an event will accomplish in time.
There are certain Christian doctrines that render the compatibility of evil and God’s existence more probable.
This last point is to say Pr(E/G&D) > Pr(E/G). Where E is amount of evil in the world, G is God's existence, and D is other Christian doctrines, the formula succinctly expresses the idea that the probability of evil and God co-existing when other Christian doctrines are true, is greater than the probability that evil and God co-exist without taking them into consideration.
William Lane Craig provides four such doctrines.[1]
that the purpose of this life is not human happiness, but the knowledge of God.
humans are in a state of rebellion against God.
God’s purposes do not cease with the grave but are eternal.
the knowledge of God is an incommensurable good.
If these are doctrines are true, and the Bible gives us good reason to believe they are true, then together they work to outweigh any intrinsic probability that the existence of God and the amount of evil in the world is unlikely. You must take in all the relevant factors when assessing probabilities.
The logical and probabilistic versions of POE are both internal critiques of the Christian worldview. That is they use premises that the Christian is already to committed to, to try to show that there is a contradiction, or just a hard-to-reconcile difficulty, with believing both ideas at once. Both these types of internal critiques have largely been abandoned in the philosophical literature. In the arena of ideas however, these types of arguments are still wielded against Christians. Though these weapons are blunt, they can still do damage to those who're unprepared to meet them.
Footnotes:
William Lane Craig and J. P. Moreland, Philosophical Foundations for a Christian Worldview, (Downers Grove, IL.; InterVarsity Press, 2003), 544-8