the random number generators are actually "pseudo random number generators," so they are not identical to the uniform distribution over all integers that people think they are, since it's not possible for computer algorithms to do 'purely' random numbers, but for all intents and purposes, and for you or I or anyone else noticing a pattern, they are as random as it gets, as long as the site is using a proper program.
In other words, Nick is correct. Good remembering, Nick. You should get retroactive bonus points for your stats class.
Independent trials, no preference for an individual number, I say put up your dream catcher the night before and ask your ouija board or psychic friend. They've got great track records.