MINI CASE 9.1
Authors such as Fischbein, Shaugnessey Green, Konold and Kahneman and Tversky have written about the difficulties that even the ‘educated' have in coping with the probabilistic world. The fact is, as a focus of mathematical study, probability is unique in that it seeks to describe and quantify a world of random events that are unpredictable and irreversible. Moreover, mathematicians have different ways in which they view probability; Classical, Frequentist and Subjective. But most fascinating are the times when the results of probability theory run contrary to our expectations and intuition. Let me give you a quick example, quoting from Darell Huff's excellent How to Take a Chance. Huff tells the story of a run on black in a Monte Carlo casino in 1913:
‘ . . . black came up a record 26 times in succession. Except for the question of the house limit, if a player had made a one-louis ($4) bet when the run started and pyramided for precisely the length of the run on black, he could have taken away 268 million dollars. What actually happened was a near-panicky rush to bet on red, beginning about the time black had come up a phenomenal 15 times ... players doubled and tripled their stakes (believing) that there was not a chance in a million of another repeat. In the end the unusual run enriched the Casino by some millions of francs'.
This tale illustrates one of the most familiar probabilistic misconceptions - the Gambler's Fallacy or Recency Bias. It is the same fallacy that makes the Coin thrower think that Tails is more likely after he has tossed three Heads in succession. There are many other biases and misconceptions in the world of probability. Another is the Representative Bias which leads one to believe that, for example, the outcome BBBGGG for a family of six children is more likely than GGGGGG because it appears to represent the ‘typical' member of the distribution more than GGGGGG, which seems ‘unusual' and hence less probable. This is analogous to the misconception that 9, 14, 29, 32, 39, 43 is more likely to be chosen as the winning outcome for the National Lottery than 1, 2, 3, 4, 5, 6.