As has been discussed in this series of postings on quantum mechanics, probability and uncertainty are key elements of the subatomic world. Einstein found this aspect of quantum mechanics particularly distasteful, making a famous statement that "God does not play dice with the universe" (the exact wording of this quote differs, depending on the source one looks at).
Physicists have come up with some creative ways of conceptualizing probability within quantum mechanics. A particle may take on one property or another, each with some probability. How the different possibilities are represented is where the ideas get, shall we say, interesting.
Three excellent sources on quantum probability include the 1959 book The Strange Story of the Quantum by Banesh Hoffmann, the 2002 book Quantum Physics: A Beginner's Guide to the Subatomic World by John Gribbin, and a June 2005 profile of Sir Roger Penrose in Discover magazine by Tim Folger.
One approach, known as the Copenhagen Interpretation, essentially says that a system shifts around between possible outcomes while not being observed, but settles upon one definite outcome upon observation. According to Gribbin:
The Interpretation says that a quantum system does not exist in a definite state until it is measured. For example, an electron traveling through the double-slit experiment does so as a spread-out wave, and does not have a precise location in space. It is only when it arrives at the detector screen that it makes a "choice" from the probabilities (like a tumbling die finally settling with three spots facing up) which causes the wave function to "collapse," as Bohr put it, onto a single point (p. 30).
(My earlier discussion of the double-slit experiment is available here.)
Hoffmann uses a different analogy:
When we flip a coin, it is neither heads nor tails until it actually lands... The state would be changed from a combination to a pure state, and it would be changed by the very act of observation... we might even imagine the coin could be twirling... The only trouble would be that we had no way of observing the actual twirling itself... the joke is that we really do not know whether the coin was twirling at all... (excerpts from pp. 161-164).
Another approach, known as the Many Worlds Interpretation, says that each possible outcome can exist in a definite state, but each would do so in a different world (or "Parallel Universe"). As described by Folger, in his profile of Penrose:
Called the many worlds interpretation, it was proposed in 1957 by Princeton University doctoral candidate Hugh Everett III. Its adherents take the laws of quantum theory at face value: Every possible quantum outcome really exists -- but in worlds parallel to our own. In one universe, Penrose is talking with me in Oxford; in another, he is watching a monster-truck rally (p. 32).
Folger notes that, "Penrose cannot believe anyone finds either the Copenhagen interpretation or the many worlds picture satisfactory" (p. 32). Penrose has his own gravity-based interpretation, which the Discover article spells out.
Another famous vehicle for examining these issues is the Schrodinger cat scenario, which I'll also leave for readers to read about on their own.
In conclusion, unlike Einstein, others are not bothered by the probabilistic foundation of quantum mechanics. Notes Hoffmann:
...probabilities are potent things -- if only they are applied to large numbers... we really place far more confidence in the certainty of probabilities than we sometimes like to admit to ourselves when thinking of them abstractly... Roulette casinos rely on probabilities for their gambling profits... (p. 175)