Belief that an event is "due" to happen: For example, "The roulette wheel has landed on red in three consecutive spins. The law of averages says it's due to land on black!" Of course, the wheel has no memory and its probabilities do not change according to past results. So even if the wheel has landed on red in ten consecutive spins, the probability that the next spin will be black is still 48.6% (assuming a fair European wheel with only one green zero: it would be exactly 50% if there were no green zero and the wheel were fair, and 47.4% for a fair American wheel with one green "0" and one green "00"). Similarly, there is no statistical basis for the belief that lottery numbers which haven't appeared recently are due to appear soon. This sort of belief is called the gambler's fallacy.
Belief that a sample's average must equal its expected value. For example, if one flips a fair coin 100 times, there is only an 8% chance that there will be exactly 50 heads.
Belief that a rare occurrence will happen given enough time. For example, "If I send my résumé to enough places, the law of averages says that someone will eventually hire me." This may actually be true assuming nonzero probabilities and that the number of trials is really large enough; the law of averages is then simply the Law of large numbers.