The Gambler's Fallacy and the law of small numbers

 

What is the count? This is a critical question because investors are often deceived by a belief in the "law of small numbers", the erroneous view that a small sample will be representative of a large sample.  If you have a small sample size, what you may expect is not close to reality. 

This view or belief in the law of small numbers often leads to the gambler's fallacy or alternatively, the "hot hand" bias. Each is somewhat different in that they adapt different conclusions, but they are both based on mistakes associated with small sample. This idea when applied to the gambler's fallacy leads to the false conclusion that streaks will even out to be closer to the representative sample. People who believe in the law of small numbers and think they know the distribution are insensitive to sample size and will assume that results will mean revert. In the case of the hot had, a person does not have an idea about the distribution and extrapolates from the small sample to a more general view.

The gambler's fallacy has also been called the Monte Carlo fallacy or the fallacy of the maturity of chances and states that if an event has occurred less frequently, it is more likely to happen again.  If someone flips five heads in a row, the fallacy will say that the probability of a tail is more likely. This thinking is wrong. Hence, there is a fallacy. The likelihood of heads or tails is still .5 regardless of the past. 

This insensitivity to the sample size will also lead to the clustering illusion where people will see streaks with random events and see non-random behavior. 

The hot hand suggests that a sports player like a basketball shooter can have a hot hand and thus should be given the ball when he has a successful sequence of baskets. Could basket players have streak shooting? Can baseball hitters have streaks? Yes, it is possible, but the more likely effect is a clustering and sampling bias. The evidence on hot hands is mixed with some statisticians finding the effect, yet given the mixed results, a conclusion that there is a bias is more likely.

Before you get too excited about some strategy or make judgments on what will happen next when faced with an event that has a small sample size, ask the question if you are engaging the flaw of small numbers.