One problem with computers though is that they can really only "simulate" randomness unless they are connected to an external source of entropy, such as a device that measure radioactive decay or listens to radio background noise. It's called a "pseudorandom number generator".
For nearly every situation, PNRGs are fine. When people are paranoid about things being "truly" random (i.e. lotteries and casino websites) some other method is used to either control the pseudorandomness or to use a natural source.
I believe the "Gambler's Fallacy" concept really only applies to the "strictly" random events in the game, by the way. Such events include the determination of what option you will get when you search an island or lagoon (i.e. finding moves,gold,and so on). After it's been determined, i.e. you will receive a pearl, a different, possibly less-than-straight-random, routine will be used.
A perfect example is attacking. As everyone knows, when you attack someone whose defense is exactly at your offense, the odds are 50% that you will succeed. That would be a "strictly" random event in my definition. If their defense is higher or lower, then it's not entirely random any more (though there is some element of randomness to it).