Many studies of the stock market have shown that the daily averages approximate a "random walk," yet thousands of stock brokers (and millions of investors) believe they can time their investments.
Finally, the entire gaming industry capitalizes on people's beliefs in streaks, either that they are due for good luck, or the slot is due for a big pay-off.
We will see more causality than is really there, and we will over interpret the meaning of random events. For example, we will see significance in three bad outcomes "bad luck comes in threes," or we will look for an insidious cause for "bad luck" such as black cats.
Not only do we fail to recognize randomness, we sometimes think things are random when they are not. For a mundane example, most people shuffle cards only a couple of times, certainly less than 5 times, thinking that is good enough to mix up the cards. But, as the casinos have found out, it takes at least 7 good shuffles to put the cards in a random order.
Research has show than people cannot distinguish random reinforcement patterns from non-random ones, creating "illusory correlations" between their response and the outcome. Misconceptions about randomness may foster an illusion of control-- that we think we are more in control of outcomes affecting us than we really are. This can lead to superstitious behavior.
However, if people are warned ahead of time that the outcomes to their responses might be randomly determined, or that the pattern might be really random, then they can, to some degree, detect random reinforcement.
Believers in the paranormal show stronger statistical illusions (inferential errors) than non-believers, so maybe belief in the paranormal results from overinterpreting chance or fluky events. But what is the causal direction? Does poor thinking predispose people towards paranormal belief, or do paranormal beliefs cause people to distort their interpretations of events?
[See Fiske and Taylor, 1991, Plous 1993, or Gilovich, 1991 for a review of the extensive heuristics and biases literature which covers many of the points listed above]