It would seem that randomness is probably a matter of degree.
To Laplace's Demon* existing in a deterministic universe there is no such thing as randomness.
However, in our specific universe as we know (or, more properly, don't know) it, satt_focusable's definition employing our ignorance of events leading up to an event is valid.
But I sense a implicit longing for fairness or impartiality in the definition of a "random" outcome. After all, which coin flip is more random, that conducted by the opposing team captain or that effected by the perceived impartial actions of a referee? The latter, simply adds more variables to the equation that must be resolved for the correct "prediction" of the outcome. Increased predictive difficulty equals increased "fairness".
If one has the infinite and all knowing intelligence of Laplace's Demon and lives in a strictly deterministic universe, there is no such thing as randomness. It follows that fairness also dies.
So, given a deterministic world with its coin flips, rolling dice, or even fluctuations in light from deep space, this merely affords us only degrees of pseudo-randomness.
satt_focusable again comes to our rescue (somewhat...as long as we are comfortable with head-in-the-sand reliance upon our own ignorance as to causal factors) and hints at the degrees of randomness when he points to probability and statistics involved in such random events. But how much faith do we have in such "randomness" of such games of chance? Given you would receive a million tax free dollars upon the one time correct call of such random events, would you choose the 1 in 2 (50% chance of a win) coin flip or that 1 in 6 (about 17%) chance of success involved with the toss of a die? Obviously the coin flip allows less "randomness" (it also is more humane--loosing an all or nothing contest involves less suffering than missing the die toss by "one": Oooohh I missed by "just" one!
What about the last depository of true "randomness", quantum mechanics? Well, God's participation in games of craps aside, satt_focusable gives us a hint as to QM attributes towards randomness:
"In a sense "randomness" is a circumstance where events before they arise can be described only through the probability distribution. This may be certainly caused by the fact that the events are "too complex to be accurately predictable." In other words, humans are too ignorant about the mechanism in which events are caused to predict those with certainty."
Schroedinger's cat has less to do with true randomness than with the observer's effect due to his actual "observation" of an event. Indeed, it is the observer's method of observation and even the fact that there is observation at all that not only affects the outcome of an event but actually precipitates said outcome. As a matter of fact, the Schroedinger's cat situation posits that no event takes place until there is an actual attempt towards observation. Schroedinger's cat's suspension between life and death and said resolution until observation speaks more towards satt_focusable's claim of human inadequenacies towards such knowledge of QM than promotion of randomness. Randomness, whether a basis for games of chance or "Free Will" seems wanting.