The Epicurean Paradox -
Is God willing to prevent evil, but not able?
Then he is not omnipotent.
Is he able, but not willing?
Then he is malevolent.
Is he both able and willing?
Then whence cometh evil?
Is he neither able nor willing?
Then why call him God?
This is an argument that many atheists use to try to deny the existence of God. It appears on many Facebook walls, and I'm sure it's floating around Pinterest as well.
It makes a lot of Christians (and Hindus, and Muslims, and Sikhs) a bit uncomfortable. When quoted by a quick and clever thinker, it can be hard to refute on the spot. It seems to fold back in on itself quite neatly, and leave few chinks by which it can be dismantled.
I have another way...put a block of C4 on it and blow the little turkey to Mars.
The problem with the Epicurean paradox is that it's a circular argument, and one that completely depends on a straw man. The circularity comes from the a priori assumption that we know how the world should work, and that our view is unquestionably correct.
The straw man is the 'God' whose motives are inferred by His actions, and which are judged by 'our' standards.
We really have to look at the question in two parts - evil arising from free human will, and 'natural evil' like cancer.
All of the monotheistic religions mentioned above grant that we have free will, and the opportunity to choose between good and evil. This, then, would imply that the tolerance of evil was a necessary part of the creation of a world in which good could also arise. Banning free will would prevent human evil, but would also prevent human goodness. If you don't have shadow, you don't have light, and therefore no way to define 'good'.
An interim step of allowing evil intent but preventing evil action is meaningless. Even a cow knows that it can't walk through a canyon wall, and it won't try.
Natural evil follows a similar line of reasoning. We have a creation that 'works'; the parts fit together. And some of those parts are rather dangerous to us. Could God have removed them? We don't know - but we do know that He set the natural laws, and it would make Creation nonsensical if He broke them on a whim.
Think of teaching a child to drive, after having been brought up in relative isolation (rural New Mexico, maybe?) - you want her to mind the speed limits and stay in her lane. If you didn't follow the rules of the road, and she had no other role models to follow, she's be terribly confused. Why rules for her, yet none for you?
Of course, you didn't set the rules of the road. They were formulated on a social plane 'above' you. So let's look at another example - house rules, like no computer games before church on Sunday.
No law against computer games, except your own...but how would your child feel on seeing you break your own rule?
It really does come down to God avoiding "do what I say, not what I do."
We have a world that allows free will, and that obey certain physical laws. And both of these allow for the presence of evil. We may not like it, but we have to accept it as a price that God decided was worth paying.
Worth paying to raise up His companions for Eternity.