Gambler's fallacy
From Dickinson College Wiki
Gambler’s Fallacy and The Law of Small Numbers
Analytical Summarization of Matthew Rabin’s paper Inference by Believers in the Law of Small Numbers
- Tversky and Kahneman started to explore the psychology behind false assumptions made from same populations in 1971. What they dubbed as, The Law of Small Numbers, explains how financial analysts can misperceive a small sample to be indicative of the entire population.
- Take for example, a study used by Tversky and Kahneman
- they told a sample of students that “the mean IQ of the population of eighth graders in a city is known to be 100”. After telling the sample group of students this, they went on to say they selected a random sample of 50 eighth graders. The first child who they tested scored a tremendously at an IQ of 150. The question asked of the students was “What do you expect the mean IQ to be for the whole sample?
- Rational Answer
- It is impossible to say for sure what the mean average would be for the students. If the sample population was chosen randomly, then through rational we must assume it is indicative of the population as a whole. However, since one student scored 150, our mystery sample is now only 49 students. Assuming these students have a mean score of 100, then once we factor back in the student who scored 150 we get:
4900 is the total points scored on the IQ tests by the 49 sample students (under our assumptions)
4900+150 (the score of student #50) = 5050
5050/50 = 101 the average IQ of the sample- Results from the Experiment
Conclusion
- Tversky and Kahneman realized that those people who thought the samples IQ would be 100 had put to much faith in “The Law of Small Numbers”. They thought that since that small sample must be a reflection of the population despite knowing one of the scores was 50 points hire. These people figured that the other 49 must somehow make up for the genius of student #50 by collective being 50 pts. below the average. Those who answered with an average of 101 used a Bayesian school of thinking, a theory developed on the basis of rational probability.
Gambler’s Fallacy:
- The Gambler’s Fallacy is closely related to the Law of Small Numbers. It shows another situation where humans ignore rational probability. In the Gambler’s Fallacy, people expect that the second outcome of a given situation must be negatively correlated to the first. Follow the example below:
- Assume an investor is certain that a financial analyst invests successfully have of the time. If in year 1 the analyst is successful than, using the gambler’s fallacy, the investor assumes he will not be in year two. Rationally this does not make sense because year 2 is independent of year 1. If the analyst is truly successful 50% of the time then he has a 50% chance of being successful in year two. Someone using the gambler’s fallacy will underestimate these odds.
