Probability

Can there be an event that is entirely random?

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This is a very difficult question, for two reasons:
1. It is difficult to say exactly what "random" means.
2. There are unresolved questions in the foundations of quantum mechanics that are relevant to your question.

Consider, for example, flipping a fair coin. This seems random, in the sense that we don't seem to be able to predict the outcome. Half the time the coin comes up heads and half the time it's tails, and we don't know which it's going to be until it lands. But in another sense, it doesn't seem random at all: If you knew the speed at which the coin was spinning, its exact position above the table, the air currents in the room, etc., then the laws of physics should allow you to predict how it will land. If you think of randomness as being about our lack of knowledge of how things are going to turn out, then the coin flip seems random. If you think of randomness as being about some sort of indeterminacy in the world, independent of our knowledge, then the coin flip doesn't seem random. It's hard to know which is the better way to think of randomness.

Perhaps the best place to look for random events is quantum mechanics. Consider a single atom of a radioactive substance. Quantum mechanics allows us to compute the probability that the atom will decay in the next second. According to some interpretations of quantum mechanics, whether or not the atom will decay in the next second is not determined by the current state of the universe, so it is random not only in the sense that we can't predict it, but also in the sense that even if we knew everything there is to know about the atom now, we still couldn't predict it. (Contrast this with the coin example, where if we knew everything about the current state of the coin, we could predict the outcome of the flip.) It was Einstein's dissatisfaction with this interpretation of quantum mechanics that led him to object that "God does not play dice with the universe." One alternative is so-called "hidden variable" interpretations of quantum mechanics, in which there is additional information unknown to us ("hidden variables") that determines, in advance, whether or not the atom will decay. On this kind of interpretation of quantum mechanics, the decay of the atom is random only in the same sense that the coin flip is random--we can't predict it, but only because we lack some knowledge about the current state of affairs. Many people believe that hidden variable interpretations of quantum mechanics have been proven to be impossible, but there is, in fact, a successful hidden variables theory that was developed by the physicist David Bohm. (By the way, there's an easy but not very satisfying way to construct a hidden variables theory: Just say that God doesn't play dice; rather, he uses a random number table.)

So is the decay of an atom random? The answer is that it may depend on how you interpret quantum mechanics, and it may also depend on how you interpret the word "random".

A very good, not very technical reference for issues in the foundations of quantum mechanics is:

J. S. Bell, Speakable and unspeakable in quantum mechanics, Cambridge University Press, 1987.

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