LESSON TWO ECONOMIC RATIONALITY Subtopic 10 – Statistical Reasoning Created by The North Carolina School of Science and Math forThe North Carolina School of Science and Math North Carolina Department of Public Instruction.North Carolina Department of Public Instruction Created by The North Carolina School of Science and Math forThe North Carolina School of Science and Math North Carolina Department of Public Instruction.North Carolina Department of Public Instruction

Today’s Menu Monty Hall Conditional probability Bayes’ Rule Statistical reasoning fallacies

The Monty Hall Problem!

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

The Monty Hall Problem! Overwhelmingly, YES!! It is to your advantage to switch your choice! To understand why, we need to understand conditional probability…

Review: Conditional Probability

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The Math Behind Monty Hall

Bayes’ Rule

The History of Bayes’ Rule Rev. Thomas Bayes (English, )

From last week’s class experiment There has been a murder in a village of a population of about 5000 people! You are a member of a jury. You know the following about a DNA test: when two samples of DNA are from the same person, the chance of a true match is When two samples of DNA are from different people, the chance of a false match is The DNA test on a suspect is positive. How likely do you think it is that the suspect owns the DNA found at the crime scene? Choice (circle one): A 75% - 100% B 50% - 75% C 25% - 50% D 0% - 25%

From last week’s class experiment There has been a murder in a village of a population of about 5000 people! You are a member of a jury. You know the following about a DNA test: when two samples of DNA are from the same person, the chance of a true match is When two samples of DNA are from different people, the chance of a false match is The DNA test on a suspect is positive. How likely do you think it is that the suspect owns the DNA found at the crime scene? Choice (circle one): A 75% - 100% Everyone else B 50% - 75% Two people C 25% - 50% One person D 0% - 25%

From last week’s class experiment There has been a murder in a village of a population of about 5000 people! You are a member of a jury. You know the following about a DNA test: when two samples of DNA are from the same person, the chance of a true match is When two samples of DNA are from different people, the chance of a false match is The DNA test on a suspect is positive. How likely do you think it is that the suspect owns the DNA found at the crime scene? Choice (circle one): A 75% - 100% B 50% - 75% C 25% - 50% D 0% - 25% 2% is the answer

Discussion: Do jury members understand probability? Implications for law, policy, justice? Example case: People vs. Collins 1968 Bayes’ rule in the medical profession.

Understanding probability A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50% of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50%, sometimes lower. For a period of 1 year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital do you think recorded more such days? A The larger hospital B The smaller hospital C About the same (that is, within 5 percent of each other)

Class results

Understanding probability Even some experienced research scientists believe in a law of small numbers (small samples are representative of the population they are drawn from). Don’t make this mistake! Correct answer: SMALLER HOSPITAL

Judgments ignore base rates “Steve is very shy and withdrawn, invariably helpful, but with little interest in people, or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.” Given this personality sketch, which of these professions do you think Steve most likely belongs to: A Farmer B Librarian C Physician D Airline pilot E Salesman

Discussion People ignore or underweight prior probabilities. What other examples can you think of where this might lead people into poor judgment?

More Class Examples “Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.” Which is the most likely alternative? A Linda is a bank teller B Linda is a bank teller and is active in the feminist movement

More Class Examples “Neurologists estimate that although most psychopaths go undiagnosed, 1-2% of the U.S. population meets the criteria for psychopathy, characterized – among other things – by lack of remorse and higher probability of criminality. Given that you know someone is a psychopath, which is the most likely alternative?” A Psychopath commits murder B Psychopath commits murder in cold blood

Review: Conditional Probability

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Bayes’ Rule