Inductive Reasoning thinking about what might, maybe, probably is true George Matthews CC 2017 

Going Beyond Proof 

Going Beyond Proof ! Generalization 

Going Beyond Proof ! Generalization What can we tell reliably about a large group based on a small sample? 

Going Beyond Proof ! Generalization What can we tell reliably about a large group based on a small sample? How do we apply generalizations to particular individuals? 

Going Beyond Proof ! Generalization What can we tell reliably about a large group based on a small sample? How do we apply generalizations to particular individuals? ! Reasoning by analogy 

Going Beyond Proof ! Generalization What can we tell reliably about a large group based on a small sample? How do we apply generalizations to particular individuals? ! Reasoning by analogy When do analogies work and when do they break down? 

Going Beyond Proof ! Generalization What can we tell reliably about a large group based on a small sample? How do we apply generalizations to particular individuals? ! Reasoning by analogy When do analogies work and when do they break down? ! Causal reasoning 

Going Beyond Proof ! Generalization What can we tell reliably about a large group based on a small sample? How do we apply generalizations to particular individuals? ! Reasoning by analogy When do analogies work and when do they break down? ! Causal reasoning How can we tell what caused what? 

Generalizations 

Generalizations ! Eggs break when you drop them. 

Generalizations ! Eggs break when you drop them. ! Shakespeare’s plays are written in blank verse. 

Generalizations ! Eggs break when you drop them. ! Shakespeare’s plays are written in blank verse. ! Water flows downhill. 

Generalizations ! Eggs break when you drop them. ! Shakespeare’s plays are written in blank verse. ! Water flows downhill. ! Shy people tend to be quiet at parties. 

Generalizations ! Eggs break when you drop them. ! Shakespeare’s plays are written in blank verse. ! Water flows downhill. ! Shy people tend to be quiet at parties. ! Spider webs are stronger than steel. 

Generalizations ! Eggs break when you drop them. ! Shakespeare’s plays are written in blank verse. ! Water flows downhill. ! Shy people tend to be quiet at parties. ! Spider webs are stronger than steel. ! Time moves in one direction only. 

Generalizations ! Eggs break when you drop them. ! Shakespeare’s plays are written in blank verse. ! Water flows downhill. ! Shy people tend to be quiet at parties. ! Spider webs are stronger than steel. ! Time moves in one direction only. ! f = m∗a 

Generalization 95 percent of the shy people I know are quiet at parties. Thus all shy people are quiet at parties. 

Generalization 95 percent of the shy people I know are quiet at parties. Thus all shy people are quiet at parties. ! Inductive arguments appeal to experience to justify generalizations. 

Generalization 95 percent of the shy people I know are quiet at parties. Thus all shy people are quiet at parties. ! Inductive arguments appeal to experience to justify generalizations. ! How can we sure that our generalizations are accurate? 

Generalization 95 percent of the shy people I know are quiet at parties. Thus all shy people are quiet at parties. ! Inductive arguments appeal to experience to justify generalizations. ! How can we sure that our generalizations are accurate? ! How do we know that the future will be like the past? 

How Not to Take a Sample 

How Not to Take a Sample ! Ask people you know. 

How Not to Take a Sample ! Ask people you know. ! Look at a few famous or clear cases and go from there. 

How Not to Take a Sample ! Ask people you know. ! Look at a few famous or clear cases and go from there. ! Send out a survey and see who responds or put a survey on your website. 

How Not to Take a Sample ! Ask people you know. ! Look at a few famous or clear cases and go from there. ! Send out a survey and see who responds or put a survey on your website. ! Ask leading or unclear questions. 

How Not to Take a Sample ! Ask people you know. ! Look at a few famous or clear cases and go from there. ! Send out a survey and see who responds or put a survey on your website. ! Ask leading or unclear questions. ! Ask for personal or demographic information first. 

Sampling Basics population sample size confidence level margin of error 

Sampling Basics population sample size confidence level margin of error 1000 100 90% ±8 

Sampling Basics population sample size confidence level margin of error 1000 100 90% ±8 1000 100 95% ±10 

Sampling Basics population sample size confidence level margin of error 1000 100 90% ±8 1000 100 95% ±10 1000 500 95% ±4 

Sampling Basics population sample size confidence level margin of error 1000 100 90% ±8 1000 100 95% ±10 1000 500 95% ±4 300M 500 95% ±5 

Sampling Basics population sample size confidence level margin of error 1000 100 90% ±8 1000 100 95% ±10 1000 500 95% ±4 300M 500 95% ±5 4000 100 95% ±10 

Sampling Basics population sample size confidence level margin of error 1000 100 90% ±8 1000 100 95% ±10 1000 500 95% ±4 300M 500 95% ±5 4000 100 95% ±10 4000 100 80% ±7 

Strong or Weak? 

Strong or Weak? I asked 100 of my closest friends, all of whom go to PSU, whether they believe that aliens are living among us, 45% said yes, 40% said no, 15% were not sure. Therefore we can be very confident that more PSU students believe that aliens are living among us than do not believe this. 

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A study of the rates of a certain form of rare cancer has found that: 

A study of the rates of a certain form of rare cancer has found that: A. Some rural counties have lower than average rates of this cancer among their residents. 

A study of the rates of a certain form of rare cancer has found that: A. Some rural counties have lower than average rates of this cancer among their residents. B. Some rural counties have higher than average rates of this cancer among their residents. 

A study of the rates of a certain form of rare cancer has found that: A. Some rural counties have lower than average rates of this cancer among their residents. B. Some rural counties have higher than average rates of this cancer among their residents. How can we explain this? 

Insensitivity to Sample Size A study of the rates of a certain form of rare cancer has found that: A. Some rural counties have lower than average rates of this cancer among their residents. B. Some rural counties have higher than average rates of this cancer among their residents. How can we explain this? 

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The Hot Hand A basketball player scores on an average 50% of her shots. Sometimes she shoots better than average, sometimes she shoots worse. Is she more likely to score after she has scored on six shots in a row? 

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Which would you pay more for: insurance against death by terrorist attack, or death by any cause? 

Dramatic Instances Effect Which would you pay more for: insurance against death by terrorist attack, or death by any cause? 

Argument by Analogy 

Argument by Analogy Rats get cancer when they smoke cigarettes. Rat bodies are analogous with human bodies. Thus cigarettes cause cancer in humans. 

Strong and Weak Analogies 

Strong and Weak Analogies Galileo was ridiculed at first, but then his ideas prevailed. The same is true of my theory of free energy, you may ridicule it now but it too will prevail. 

Strong and Weak Analogies Newton showed how the movement of the moon around the earth is analogous with the movement of cannon balls on earth – both result from the combined effects of inertial motion and gravitational acceleration. 

Strong and Weak Analogies Darwinian evolution operates by means of “natural selection” which is just like selective breeding of domestic animals and plants except that there is no breeder. 

Strong and Weak Analogies Just like historically important events have wide-reaching consequences, so their causes must be equally significant. Hence JFK could not possibly have been shot by “one lone nut” but there must have a been a massive conspiracy to assassinate him and a cover-up afterwards. 

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Mary was a finance major who was an outspoken advocate for women’s rights in college. Which is more likely to be true about her? 

Mary was a finance major who was an outspoken advocate for women’s rights in college. Which is more likely to be true about her? A. She works at a bank. 

Mary was a finance major who was an outspoken advocate for women’s rights in college. Which is more likely to be true about her? A. She works at a bank. B. She is a feminist who works at a bank. 

Conjunction Fallacy Mary was a finance major who was an outspoken advocate for women’s rights in college. Which is more likely to be true about her? A. She works at a bank. B. She is a feminist who works at a bank. 

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Suppose I flip a fair coin and get 5 heads in a row. Which is more likely on the next flip, heads or tails? 

Gambler’s Fallacy Suppose I flip a fair coin and get 5 heads in a row. Which is more likely on the next flip, heads or tails? 

Whenever I have a bad cold I drink a bottle of whiskey and then in a day or two I feel better. Thus whiskey cures colds. 

False Cause Fallacy Whenever I have a bad cold I drink a bottle of whiskey and then in a day or two I feel better. Thus whiskey cures colds. 

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When the players on the team do really well and I praise them for their efforts, they do less well afterward. But when they do poorly and I yell at them, their playing soon improves. 

When the players on the team do really well and I praise them for their efforts, they do less well afterward. But when they do poorly and I yell at them, their playing soon improves. What does this show about positive and negative reinforcement? 

Base Rate Fallacy When the players on the team do really well and I praise them for their efforts, they do less well afterward. But when they do poorly and I yell at them, their playing soon improves. What does this show about positive and negative reinforcement? 

Illusory Correlation 

Illusory Correlation Subjects were given sentences to read about the behavior of two groups, identified only as group A and group B. 

Illusory Correlation Subjects were given sentences to read about the behavior of two groups, identified only as group A and group B. behaviors group A group B total desirable 18 (69%) 9 (69%) 27 undesirable 8 (30%) 4 (30%) 12 total 26 13 39 Hamilton, D; Gifford, R (1976). 

Illusory Correlation Subjects were given sentences to read about the behavior of two groups, identified only as group A and group B. behaviors group A group B total desirable 18 (69%) 9 (69%) 27 undesirable 8 (30%) 4 (30%) 12 total 26 13 39 Test subjects asked about these groups associated group B more strongly with undesirable behaviors. Why? Hamilton, D; Gifford, R (1976). 

Monty Hall Problem 

Monty Hall Problem Suppose you’re on a game show, and you’re given the choice of three doors: 

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. 

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 door 1. 

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 door 1. ! The host, who knows what’s behind the doors, opens door 3, revealing a goat. 

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 door 1. ! The host, who knows what’s behind the doors, opens door 3, revealing a goat. Should you switch to door 2?