Hacking+-+An_Introduction_to_Probability_and_Inductive_Logic

An Introduction to Probability and Inductive Logic

Overview of the Book

• Author: Ian Hacking, University Professor in Philosophy at the University of Toronto and a Professor at the Collège de France in Paris.
• Purpose: Introductory textbook on probability and induction.
• Target Audience: Designed for philosophy courses but accessible to students in social sciences (psychology, economics, political science, sociology) and medical sciences (epidemiology).
• Prerequisites: Assumes no formal training in elementary symbolic logic.
• Coverage: Comprehensive course covering basic definitions of induction and probability, including topics such as decision theory, Bayesianism, frequency ideas, and the philosophical problem of induction.
• Key Features:
  • Lively and vigorous prose style.
  • Lucid and systematic organization and presentation of ideas.
  • Many practical applications.
  • Rich supply of exercises from various fields (psychology, ecology, economics, bioethics, engineering, political science).
  • Numerous brief historical accounts of the development of fundamental ideas.
  • Full bibliography for further reading.

Publishing Information

• Publisher: Cambridge University Press.
• First Published: 2001.
• Copyright: © Ian Hacking 2001.
• ISBN: $0-521-77287-7$ (hardback), $0-521-77501-9$ (paperback).

Contents Overview

• Logic: Chapters $1$ and $2$ introduce logic and inductive logic.
• How to Calculate Probabilities: Chapters $3$ through $7$ cover fundamental probability calculations, including conditional probability and Bayes' Rule.
• How to Combine Probabilities and Utilities: Chapters $8$ through $10$ delve into expected value and decision-making under uncertainty.
• Kinds of Probability: Chapters $11$ and $12$ explore different philosophical theories and meanings of probability.
• Probability as a Measure of Belief: Chapters $13$ through $15$ focus on personal probabilities, coherence, and learning from experience through a Bayesian lens.
• Probability as Frequency: Chapters $16$ through $19$ examine stability, normal approximations, significance, power, confidence, and inductive behavior.
• Probability Applied to Philosophy: Chapters $20$ through $22$ address the philosophical problem of induction and various evasions or solutions.

Cover Illustration Note: "The Allegory of Fortune" by Dosso Dossi (1486-1542)

• Figures: Young woman on the right is Fortuna (Lady Luck); young man on the left is Chance.
• Symbolism of Fortuna: Holds an enormous bunch of fruits (good luck), has only one sandal (can bring bad luck), and sits on a soap bubble (luck does not last).
• Symbolism of Chance: Holding lottery tickets. Dossi, a court painter near Venice, likely depicted this reflecting Venice's recent introduction of a state lottery.
• Artist's Belief: Art critics suggest Dossi believed life is a lottery for everyone.
• Location: J. Paul Getty Museum, Los Angeles.

Foreword: Introduction to Inductive Logic

Inductive vs. Deductive Logic

• Deductive Logic: With true premises and a valid argument, the conclusion must be true. Deductive arguments are risk-free.
• Inductive Logic: Takes risks. True premises and a good argument can still lead to a false conclusion. Uses probability to analyze risky arguments.

Importance of Inductive Reasoning

• Plays a much larger part in everyday life than deductive reasoning, guiding constant risky decisions.

Human Challenges with Risk

• People are often poor at reasoning about risks and make many mistakes when using probabilities.
• Odd Questions: The book begins with seven "Odd Questions" to illustrate common errors in probabilistic reasoning. These questions revealed that children sometimes perform better than professors.

Practical Aims of the Book

• To help readers understand, use, and act on probabilities, risks, and statistics.
• To navigate situations involving uncertainty, drawing inferences from inconclusive data, and making decisions when unsure.
• To understand how probabilities and statistics are used, abused, and how people can be fooled by numbers or when numbers conceal ignorance.

Ubiquity of Probabilities and Statistics

• Modern life is saturated with probabilities, statistics, and risk across all domains (jobs, health, environment, politics, etc.).
• This is a relatively new phenomenon compared to earlier generations.

Philosophical Aspects

• Problem of Induction: A famous philosophical problem discussed at the end of the book.
• Ethical Questions: About risk, such as maximizing common good vs. duty-based ethics (Chapter $9$).
• Religious Belief: Probability arguments for and against religious belief (Chapter $10$).
• Nature of Probability Itself: Philosophical arguments about the basic ideas of inductive inference, with the book presenting competing schools of thought rather than pretending there are none.

Calculation and Ideas

• The book focuses on the ideas behind calculations rather than precise computational solutions.
• Minimal use of pocket calculators is encouraged for exercises, as numbers often "cancel" for easy mental solutions.

Gambling as a Model

• Many simple probability examples involve games of chance, which can be controversial.
• Author's Stance: The book is not an advertisement for gambling; rather, it views regular gambling as a "waste of time, money, and human dignity."
• Utility of Gambling Models: In everyday life, people "gamble" by making decisions under uncertainty or drawing inferences from inconclusive data. Gambling models help clarify these types of decisions and inferences, connecting to practical issues like courtroom testimony and medical diagnosis.

Odd Questions (Preview)

$1$. Births in Hospitals

• Scenario: City General (large hospital) vs. Cornwall (small hospital).
• Definition: Normal week (45$%-55$$55192.

2. Pia's Profile

• Scenario: Pia, 31, single, outspoken, smart, philosophy major, activist.
• Question: Rank six statements about Pia in order of probability (most to least).
• Statements: (a) Active feminist, (b) Bank teller, (c) Works in small bookstore, (d) Bank teller and active feminist, (e) Bank teller, active feminist, takes yoga, (f) Works in small bookstore, active feminist, takes yoga.
• Discussion Point: Page 65.

3. Lotto 6/49 Tickets

• Scenario: Choosing between two lottery tickets (A: 1,2,3,4,5,6$vs. B:$39,36,32,21,14,3).
• Question: Prefer A, B, or indifferent?
• Discussion Point: Page 30.

4. Rolling Dice

• Scenario: Throwing dice to get a total of 6$or$7.
• Question: Expect to throw 7$more frequently,$6$more frequently, or$6$and$7 equally often?
• Answer Options: (a) 7$more frequently, (b)$6 more frequently, (c) Equally often.
• Discussion Point: Page 43.

5. Taxicab Accident

• Scenario: Town with Green Cabs (85$$15$$800.8$, (b) More likely blue, but probability less than$0.8, (c) Equally probable green as blue, (d) More likely green than blue.
• Discussion Point: Page 72.

6. Strep Throat Diagnosis

• Scenario: Physician suspects strep throat. Test results (not perfect): 70$$9076.

7. Imitating a Coin Toss Sequence

• Scenario: Write down a 100 H/T sequence to fool others into thinking it's from a fair coin.
• Discussion Point: Page 30.

1 Logic

Purpose of Logic

• Logic is about good and bad reasoning.
• Logicians provide precise meanings for ordinary words to discuss reasoning clearly.

Arguments

• Ordinary Language Definitions:
  • 1. A quarrel.
  • 2. A discussion with reasons for/against a proposition.
  • 3. A point or series of reasons supporting a proposition (conclusion).
• Logician's Definition: Logicians use definition (3). Reasoning is stated or written as arguments.
• Structure of an Argument:
  • Premises: A point or series of reasons.
  • Conclusion: The proposition supported by the premises.
• Propositions: Both premises and conclusions are propositions—statements that can be either true or false.

Going Wrong with Arguments

• Arguments are convincing when premises are true and provide good reasons for the conclusion.
• Two Ways an Argument Can Go Wrong:
  • The premises may be false.
  • The premises may not provide a good reason for the conclusion.
• Example: Argument (*J)
  • Premises: "If James wants a job, then he will get a haircut tomorrow." "James will get a haircut tomorrow."
  • Conclusion: "James wants a job."
  • This is a bad argument because premises could be true and conclusion false (e.g., James has a date, family visit, or routine haircut).
  • Fallacy: Argument (*J) commits a fallacy, a common error in reasoning.
  • Affirming the Consequent: The first premise is "If A, then C" (A=antecedent, C=consequent). The second premise is "C." The conclusion is "A." Inferring A from C is the fallacy of affirming the consequent.

Two Ways to Criticize Any Argument

• Example: Argument (J)
  • Premises: "If James wants a job, then he will get a haircut tomorrow." "James wants a job."
  • Conclusion: "James will get a haircut tomorrow."
  • Here, premises do provide a conclusive reason, but they could still be questioned (e.g., James wants to be a rock musician, or doesn't actually want a job).
• Basic Criticisms:
  • Challenge the premises (show at least one is false).
  • Challenge the reasoning (show premises are not a good reason for the conclusion).
• Logic's Focus: Logic is concerned only with whether the reasoning is good or bad, not generally with the truth or falsity of premises.

Validity

• Example: Argument (K)
  • Premises: "Every automobile sold by Queen Street Motors is rust-proofed." "Barbara's car was sold by Queen Street Motors."
  • Conclusion: "Barbara's car is rust-proofed."
  • If premises of (K) are true, conclusion must be true.
• Definition of Valid Argument: It is logically impossible for the conclusion to be false, given that the premises are true.
• Logical Form: Validity is best explained by logical form. Forms like (J) and (K) are valid:
  • 1. If A, then C.
  • 2. A.
  • So: 3. C.
  • 4. Every F is G.
  • 5. b is F.
  • Therefore: 6. b is G.
• Technical Term: "Valid" is a technical term in deductive logic. The opposite is "invalid."
• Example of Invalidity: Argument (*K)
  • Premises: "Every automobile sold by Queen Street Motors is rust-proofed." "Barbara's car is rust-proofed."
  • Conclusion: "Barbara's car was sold by Queen Street Motors."
  • Invalid because the conclusion could be false even with true premises (e.g., Barbara bought her car elsewhere).

True Versus Valid

• Propositions: Are true or false.
• Arguments: Are valid or invalid.
• Distinction: Do not confuse the argument (K) with a conditional proposition (e.g., "If Barbara's car was sold by QSM, and if every automobile sold by QSM is rust-proofed, then Barbara's car is rust-proofed.").
• Correspondence: An argument is valid if and only if its corresponding conditional proposition is a truth of logic.

Metaphors for Validity

• The conclusion follows from the premises.
• Whenever premises are true, the conclusion must be true.
• The conclusion is a logical consequence/implicitly contained in the premises.
• Valid argument forms are "truth-preserving" (start with true premises, end with a true conclusion).
• Core Idea: Valid arguments are risk-free arguments.

Soundness

• Definition: An argument is sound when all premises are true AND the argument is valid.
• Reasons for Unsoundness:
  • A premise is false.
  • The argument is invalid.
• Focus: Validity is about logical connection; soundness concerns both validity and the truth of premises.
• Analogy: Building a House:
  • Built on sand (false premises) ➡️ falls down.
  • Badly built (invalid argument) ➡️ falls down.
  • Built on sand and badly designed ➡️ might still stand (invalid argument, false premises, true conclusion).
• Criticisms of a Deduction: (1) A premise is false, (2) The argument is invalid.

Validity Is Not Truth!

• A valid argument can have a false premise but a true conclusion.
  • Example (R): "Every famous philosopher who lived to be over ninety was a mathematical logician." "Bertrand Russell was a famous philosopher who lived to be over ninety." So: "Bertrand Russell was a mathematical logician."
  • Valid, true conclusion, but first premise is false (e.g., Thomas Hobbes).

Invalidity Is Not Falsehood!

• An invalid argument can have true premises and a true conclusion.
  • Example (*R): "Some philosophers now dead were witty and wrote many books." "Bertrand Russell was a philosopher, now dead." So: "Bertrand Russell was witty and wrote many books."
  • True premises, true conclusion, but invalid.

Division of Labor

• Experts on Truth of Premises: Detectives, nurses, pollsters, historians, ordinary people.
• Experts on Validity: Logicians (they study premise-conclusion relations, not empirical truth).

2 What Is Inductive Logic?

• Inductive logic focuses on risky arguments, analyzing them using probability.
• A good risky argument can have true premises but a false conclusion.

Examples of Risky Arguments

• The Big Bang theory is well-supported but could be wrong.
• Smoking causes lung cancer is strongly evidenced, but the reasoning is still risky (e.g., common predisposition).
• A company marketing a left-handed mouse takes a risk on profitability.
• Assuming a friend will take a logic class for social reasons is a risky argument.

Oranges (Sample-to-Population)

• Context: Buying a box of oranges, seeking good ones at half-price.
• Argument (A): "This orange is good." ➡️ "All (or almost all) the oranges in the box are good."
  • Risky: One orange is not very good evidence.
• Argument (B): "This orange that I chose at random is good." ➡️ "All (or almost all) the oranges in the box are good."
  • Risky, but less risky than (A) due to random selection.
• Argument (C): "Of these six oranges that I chose at random, five are good and one is rotten." ➡️ "Most (but not all) of the oranges in the box are good."
  • Risky, but based on more data than (B). Still, chance could lead to a misleading sample.

Samples and Populations: Basic Forms of Risky Arguments

• Sample to Population: Statement about a sample ➡️ Statement about the population as a whole (e.g., arguments A-C).
• Population to Sample: Statement about a population ➡️ Statement about a sample (e.g., knowing most oranges are good, inferring my four chosen oranges are good).
• Sample to New Sample: Statement about a sample ➡️ Statement about a new sample (e.g., four random oranges are good, so the next four will be good).

Proportions

• Risky arguments can be made more exact by using proportions (e.g., 90904$is$1/6."
  • Valid: Conclusion must be true if premises are true and basic laws of probability (sum to 1 for mutually exclusive, exhaustive events) hold.
• Example 2 (Fair Die): "This die has six faces… Each face is equally probable." ➡️ "The probability of rolling a 3$or a$4$is$1/3."
  • Valid: Conclusion must be true given premises and basic laws (probabilities add for mutually exclusive events).
• Contrast (Risky Die Argument): "This die has six faces… In 227$rolls, a$4$was rolled just$38$times." ➡️ "The probability of rolling a$4$with this die is about$1/6."
  • Risky: Conclusion might be false, even with true premises (e.g., die is biased against 4 but got lucky in these rolls).

Another Kind of Risky Argument: Inference to the Best Explanation

• Risky arguments don't always involve probability or sample-to-population reasoning.
• Example (Grades): "Almost all the students in that class got As." ➡️ "The instructor must be a really easy marker."
  • This offers a hypothesis to explain observed facts. Other explanations are possible (gifted class, marvelous teacher, easy material).
• Example (James's Haircut, revisited): Argument (*J)$from Chapter$1 can be seen as inferring the most plausible explanation (James wants a job) given certain details (he rarely cuts his hair, is broke, is getting a cut tomorrow).
• Definition: Inference to the best explanation is when one explanation is much more plausible than any other.
  • Often used in science (e.g., Big Bang theory and background radiation).

"Abduction" (Charles Sanders Peirce)

• Peirce's Categories of Argument (groups of three):
  • Deduction.
  • Induction (risky).
  • Abduction (Peirce's term for inference to the best explanation, distinct from induction).
• Debate: Some philosophers use probability for abduction; Peirce did not.
• Book's Scope: This book will not discuss inference to the best explanation (abduction).

Testimony

• Most beliefs are based on testimony (e.g., parents, instructors, news).
• Examples:
  • "My mother told me I was born on Feb. 14$th." ➡️ "I was born on Feb. 14$th."
  • "My psychology instructor says Freud was a fraud." ➡️ "Freud is a worthless guide."
  • "Evening news says mayor is meeting officials about the flood." ➡️ "The mayor is meeting officials."
• Risky: All these are risky arguments (e.g., mother might have lied, news might be misinformed).
• Debate: Some testimony can be analyzed with probability, but many problems remain.
• Book's Scope: This book will not discuss testimony in detail, focusing mainly on probability.

Rough Definition of Inductive Logic

• Inductive logic analyzes risky arguments using probability ideas.

Decision Theory

• Concerns reasoning about what to do, not just what to believe.
• Ingredients for Decision:
  • What will probably happen (beliefs, measured by probabilities).
  • What we want (values, measured by utilities).
• Rough Definition of Decision Theory:
  • Analyzes risky decision-making using ideas of probability and utility.

3 The Gambler's Fallacy

• Main Ideas: Independence, randomness, probability model.

Roulette Scenario

• Wheel: 38 segments (18 black, 18 red, 2 green/zeros).
• Bet: 10$on red, wins$20. Loses on black/zero.
• Gambler's Situation: Long run of 12 black spins. Gambler bets on red, reasoning: "The wheel must come up red soon" because it's fair and reds/blacks should occur equally often, so it needs to "even out."
• Conclusion: This is an argument and a risky decision.
• Problem: This chapter is called "the gambler's fallacy," implying the gambler's argument is flawed.

"Fairness"

• Meaning of Fair: In gambling, "fair" means unbiased.
• Biased: A coin tends to come up heads more often; a roulette wheel tends to be red more often.
• "Tends": Refers to "on average" or "in the long run."
• Chance Setups: Systems involving chance, allowing repeated trials (tosses, spins, draws).
• Outcomes: Definite set of possible outcomes for each trial (e.g., heads/tails, 1-6 for a die, 38 segments for roulette).
• Unbiased Setup: Each outcome occurs as often as every other in the long run.

Independence

• Lack of bias alone doesn't guarantee fairness.
• Example (Coin Tossing Trick): A coin can be tossed to alternate H/T regularly. It's unbiased (H and T occur equally often) but not "fair" because outcomes are not random.
• Randomness: Outcomes are not influenced by previous trials; the setup has no "memory."
  • Sometimes defined as: no successful gambling system is possible.
  • Related to complexity: random sequences are too complex to predict, measured by the length of the shortest computer program to generate them.
• Definition of Independent Trials: Trials on a chance setup are independent if and only if the probabilities of the outcomes of a trial are not influenced by the outcomes of previous trials.
• Definition of Fair Chance Setup: Unbiased and outcomes are independent of each other.

Two Ways to Be Unfair

• Bias: Tends to favor one outcome over others.
• Lack of Independence: Regularity in outcomes, past outcomes influence future ones.
• Four Combinations:
  • Fair: Unbiased, independent.
  • Unfair: Unbiased, not independent (e.g., controlled coin toss, sampling without replacement).
  • Unfair: Biased, independent (e.g., loaded dice, some knucklebones).
  • Unfair: Biased, not independent (e.g., urn with more red balls, sampled without replacement).

The Gambler's Fallacy

• Core of the Fallacy: Involves a misunderstanding of independence, not bias.
• Gambler's Reasoning: Believes the fair roulette wheel (unbiased, independent) must produce red soon after a run of 12 blacks to "even out."
• Inconsistency: The gambler assumes fairness (so, independence) but then infers that past results (12 blacks) affect future outcomes (red is more likely), which contradicts independence. Thus, the gambler is inconsistent.

Impossibility of a Successful Gambling System

• Outcomes are random if no betting system is guaranteed to win.
• If trials are not independent (e.g., coin never produces HTH or THT), a profitable system is possible.
• The fallacious gambler believes his system ("bet red after 12 blacks") will work, implying non-independence, which contradicts his premise of a fair wheel.

Compound Outcomes

• For a fair setup, all sequences of outcomes are equally probable (e.g., HHTT and HHHH).
• A sequence of 12 blacks is followed by red half the time and black half the time.

Odd Question 3: Lotto 6/49 (revisited)

• Scenario: Choose between ticket A (1,2,3,4,5,6) and B (random-looking numbers).
• Indifference: If the lottery is fair, all sequences are equally probable, so you should be indifferent purely by probability.
• Pragmatic Choice: Choosing A might be advantageous because fewer people choose regular-looking sequences. If A wins, the prize might be larger for each winner as it's split among fewer people.
• Counter-argument: Many people know this, leading to more choosing A, which would reduce individual winnings.

Odd Question 7: Imitating a Coin (revisited)

• Task: Write a 100 H/T sequence that appears random.
• Common Error: Most people try to make sequences too irregular, avoiding long runs.
• Reality: In 100 tosses of a fair coin, a run of 7$heads in a row is more probable than not, and a run of$6 heads in a row is almost certain.
• Relevance to Gambler's Fallacy: People feel runs have to even out quickly. But with independent trials, the coin has "all the time in the world" for things to even out.

The Alert Learner

• Stodgy Logic: Points out the gambler's inconsistency (assuming fairness while expecting non-independence).
• Alert Learner's Reasoning: Observes 12 consecutive blacks and infers the wheel must be biased towards black, betting on black.
• Nature of the Argument: Alert Learner makes an inductive argument to the risky conclusion that the wheel is biased.
• Real-world Context: Pure logic can't resolve Alert Learner's claim; it depends on external knowledge (casino practices).
  • In a real casino, a bias is unlikely as it would be quickly exploited.

Risky Airplanes

• Scenario: Two airlines, Alpha Air and Gamma Goways. Gamma just crashed a plane.
• Vincent (Gambler's Fallacy-like): Takes Gamma, reasoning they crash only "one in a million" and just had their crash, implying it's less likely to crash again soon.
• Gina (Alert Learner-like): Takes Alpha, reasoning Gamma is negligent due to the accident.
• Other Perspectives: Grandma thinks Gamma will be extra careful; boyfriend points to Gamma's safety violations.
• Conclusion: Different premises lead to different risky conclusions. Logic identifies Vincent's reasoning as fallacious if he assumes independence but expects the rate to "even out."

Models

• Purpose: Simple, artificial models (dice, cards, urns) are used to understand complex real-life situations.
• Benefit: Allow application of precise, often mathematical/logical concepts.
• Challenge: Determining the applicability of the model to the real-life problem.
• Example (Parking): Modeling parking ticket risk with a lottery (red/green cards).
  • If inspectors never patrol two nights in a row, then the model changes to dependent trials.
• Question: What is the right model for Gamma Goways?

Two Ways to Go Wrong with a Model

• Error 1: The model may not represent reality well (mistake about the real world).
  • Alert Learner's criticism: The initial model of a fair wheel was wrong.
• Error 2: Drawing wrong conclusions from the model (logical error).
  • Fallacious Gambler's error: Forgot independence within his fair-wheel model.
• Connection to Argument Criticism: Criticizing the model is like challenging premises; criticizing analysis of the model is like challenging reasoning.

4 Elementary Probability Ideas

What Has a Probability?

• Car Insurance Example:
  • "Probability that you will have an automobile accident next year." (Asks about a proposition.)
  • "Probability of your having an automobile accident next year." (Asks about an event.)
• These are two ways of asking the same question.

Propositions and Events

• Logician's Focus: Arguments, premises, conclusions = propositions (true or false).
• Statistician's Focus: Events (occur or don't occur).
• Languages: Two languages of probability. Most statements can be translated between them.
• Why Learn Two Languages?: Accommodates different student backgrounds, future studies, and promotes understanding across disciplines.
  • Emphasizes clarity and adaptability in expressing ideas.

Notation: Sets (Statistician's Language)

• Disjunction (AvB): Union of sets of events (AUB).
• Conjunction (A&B): Intersection of sets of events (AB).
• Negation (~A): Complement of a set of events (A').

Notation: Logic (General Language)

• Capital Letters (A, B, C…): Represent propositions or events.
• Disjunction (AvB): "A, or B, or both" (read "A or B").
• Conjunction (A&B): "A and B."
• Negation (~A): "Not A."
• Example (Roulette): Z (zero), B (black).
  • ZvB = wheel stops at zero or black.
  • ~R (not red) is equivalent to (ZvB).

Notation: Probability

• Pr( ): Notation for probability (e.g., Pr(Z), Pr(ZvB), Pr(~(ZvB)), Pr(R)).
• Since ~R is equivalent to (ZvB), Pr(~R) = Pr(ZvB).

Two Conventions

• Probabilities between 0$and$1$</strong>:$0 \le \text{Pr(A)} \le 1.
• Certainty: Probability of what is certainly true or bound to happen is (1).
  • Sure event/proposition (omega, \Omega$) has$\text{Pr}(\Omega) = 1.

Mutually Exclusive

• Definition: Two propositions/events are mutually exclusive (disjoint, incompatible) if they cannot both be true/occur at once.
  • Example: Roulette wheel stopping at zero AND stopping at red on the same spin.

Adding Probabilities

• Rule: If A and B are mutually exclusive, then \text{Pr(AvB)} = \text{Pr(A)} + \text{Pr(B)}.
  • Example: Roulette Pr(Z)=1/19$, Pr(R)=$9/19$. Pr(ZvR) =$1/19 + 9/19 = 10/19.
  • Example (Fair Die): Pr(Even) = 1/2$(Pr(2)+Pr(4)+Pr(6) =$1/6+1/6+1/6).
• Caution (Overlap): Cannot simply add if events overlap.
  • Example: E = even, M = ace or prime (1,2,3,5).
  • Pr(E)=3/6$, Pr(M)=$4/6 (not mutually exclusive, 2 is in both).
  • \text{Pr(EvM)} \neq \text{Pr(E)} + \text{Pr(M)}$($7/6 is impossible).
  • Actually, EvM = (1,2,3,4,5,6), so Pr(EvM) = 1.
• Summary: Adding probabilities is for mutually exclusive events/propositions.

Independence

• Intuitive Definition: Two events/propositions are independent when the occurrence/truth of one does not influence the probability of the other.
• Still, many people (like Fallacious Gambler) misunderstand independence.

Multiplying Probabilities

• Rule: If A and B are independent, then \text{Pr(A&B)} = \text{Pr(A)} \times \text{Pr(B)}.
  • Example (Two Fair Dice): Pr(\text{Five}1 & Six2$) = Pr($\text{Five}1$)$\times$Pr($\text{Six}2$) =$1/6 \times 1/6 = 1/36.
• Caution (Non-Independence): Cannot simply multiply if events are not independent.
  • Example (Fair Die): Pr(E)=1/2$, Pr(M)=$2/3. Pr(E&M) \neq 1/2 \times 2/3 = 1/3, because 2 is the only outcome that is both even and prime. So Pr(E&M) = Pr(2) = 1/6.
  • Example (Blue Jays & Dodgers): Pr(D&J) \neq$Pr(D)$\times Pr(J) if their success might be interdependent (e.g., player trades).
• Summary: Multiplying probabilities is for independent events/propositions.

Sixes and Sevens: Odd Question 4 (revisited)

• Question: Is 6$or$7 more probable with two fair dice?
• Common Mistake: Many think they're equally probable.
• Correct Answer: 7 is more probable.
• Analysis (Two Fair Dice, 36 outcomes):
  • Sum of 7$:$(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$. There are$6 ways.
  • Pr(7$with$2$dice) =$6/36 = 1/6.
  • Sum of 6$:$(1,5), (2,4), (3,3), (4,2), (5,1)$. There are$5 ways.
  • Pr(6$with$2$dice) =$5/36.

Compounding Probabilities

• Involves combining independent and mutually exclusive events.
• Example (Coin + Urns):
  • Urn 1: 3 Red, 1 Green (Pr(R1) = 3/4).
  • Urn 2: 1 Red, 3 Green (Pr(R2) = 1/4).
  • Fair coin toss: Heads (Urn 1, Pr(H)=1/2$), Tails (Urn 2, Pr(T)=$1/2).
  • Pr(H&R1) = Pr(H) \times$Pr(R1) =$1/2 \times 3/4 = 3/8.
  • Pr(T&R2) = Pr(T) \times$Pr(R2) =$1/2 \times 1/4 = 1/8.
  • Pr(Red) = Pr(H&R1) + Pr(T&R2) (mutually exclusive) = 3/8 + 1/8 = 1/2.

A Trick Question (Laplace's Favorite)

• Scenario (Coin + Urns): Pick an urn, then draw two balls with replacement.
• Question: Pr(two reds in a row)?
• Common Mistake: Thinking Pr(Red) \times$Pr(Red) =$1/2 \times 1/2 = 1/4.
• Correct Analysis:
  • Event X: Toss Heads, then R from Urn 1, then R from Urn 1 (Pr(X) = 1/2 \times 3/4 \times 3/4 = 9/32).
  • Event Y: Toss Tails, then R from Urn 2, then R from Urn 2 (Pr(Y) = 1/2 \times 1/4 \times 1/4 = 1/32).
  • Pr(two reds) = Pr(X) + Pr(Y) = 9/32 + 1/32 = 10/32 = 5/16.
• Explanation: Once an urn is picked (introducing a bias for that draw), it's more probable that both balls will be of the majority color from that chosen urn.

Laplace (1749-1827)

• P. S. de Laplace: Major figure in probability theory, wrote the first introductory college textbook ("A Philosophical Essay on Probabilities").
• Also made foundational contributions to mathematics and astronomy.

5 Conditional Probability

• Most important new idea about probability.
• Probability of something happening on condition that something else happens.

Categorical and Conditional

• Categorical Probability: "The probability that Stefan will beat Boris is 4050-50 chance that Stefan would beat Boris." (Probability given a condition).
  • Examples:
    • Bumper grain crop given heavy snowfall.
    • Dealing an ace as the second card given the first was a king (4/51$) or an ace ($3/51).
• Bingo Analogy: Players get excited as they fill a line because the conditional probability of winning increases.

Parking Tickets Analogy

• Categorical: Pr(ticket) \approx 1/7.
• Conditional: Pr(ticket tonight | ticket last night) = 0 (if not patrolled two consecutive nights).

Definition of Conditional Probability

• Notation: Categorical: Pr( ). Conditional: Pr( / ).
• Formula: If Pr(B) > 0$, then$\text{Pr(A/B)} = \operatorname{Pr(A\&B)} / \operatorname{Pr(B)}.
  • Pr(B) must be positive to avoid division by zero.

Conditional Dice Example

• Scenario: Fair die. Pr(6 | Even)?
  • Intuitive: 1/3$(out of${2,4,6}).
• Using Definition:
  • Pr(6 & Even) = Pr(6) = 1/6.
  • Pr(Even) = 1/2.
  • Pr(6 | Even) = (1/6) / (1/2) = 1/3. (Matches intuition).

Overlapping Outcomes Example

• Scenario: Fair die. E = even, M = Ace or Prime (1,2,3,5). Pr(Even | M)?
  • Only common outcome is 2.
• Using Definition:
  • Pr(Even & M) = Pr(2) = 1/6.
  • Pr(M) = 4/6.
  • Pr(Even | M) = (1/6) / (4/6) = 1/4.

Well-Shuffled Cards Example

• Scenario: Top card dealt from 52-card deck. Know it's Red or Clubs (RvC).
• Question: Pr(Ace | RvC)?
• Analysis:
  • RvC includes 26 red cards + 13 clubs = 39 cards.
  • A & (RvC) = Ace of clubs, or a red Ace (diamonds, hearts) = 3 aces.
  • Conditional probability = 3/39 = 1/13.
• Using Definition:
  • Pr(A & (RvC)) = 3/52.
  • Pr(RvC) = 39/52.
  • Pr(A | RvC) = (3/52) / (39/52) = 3/39 = 1/13.

Urn Example (using conditional probability)

• Scenario: Urn A (80% R, 20% G), Urn B (40% R, 60% G). Pick an urn at random (Pr(A)=Pr(B)=0.5). Replace ball after draw.
• Given: You draw a red ball (R).
• Question: Pr(A | R)? (If you're like Alert Learner, anticipate higher chance for Urn A).
• Solution Steps:
  1. Pr(R/A) = 0.8$, Pr(R/B) =$0.4.
  2. Pr(A&R) = Pr(R/A)Pr(A) = 0.8 \times 0.5 = 0.4.
  3. Pr(B&R) = Pr(R/B)Pr(B) = 0.4 \times 0.5 = 0.2.
  4. Pr(R) = Pr(A&R) + Pr(B&R) (A&R and B&R are mutually exclusive) = 0.4 + 0.2 = 0.6.
  5. Pr(A | R) = Pr(A&R) / Pr(R) = 0.4 / 0.6 = 2/3.

Drawing the Calculation to Check It: Tree Diagram

• Visualizes the paths to a red ball. First branch (0.5 for A, then 0.8 for R from A gives 0.4). Second branch (0.5 for B, then 0.4 for R from B gives 0.2). Sum is 0.6.
• Pr(A | R) is the probability of the A-branch leading to R, divided by total Pr(R) = 0.4 / 0.6 = 2/3.

Models (beyond urns)

• Applying standard models to real-life situations.

Shock Absorbers Example

• Scenario: Bolt supplies 40$$6096$$0.96).
• Bolt shocks: 72$$0.72).

Weightlifters Example

• Scenario: Steroid team (80% users), Cleaner team (20% users). Coach flips fair coin to choose team.
• Given: One randomly tested member (U) uses steroids.
• Question: Pr(Steroid team | user)? (Pr(S | U)?)
• Solution:
  • Pr(S) = Pr(C) = 0.5.
  • Pr(U/S) = 0.8$, Pr(U/C) =$0.2.
  • Pr(S&U) = 0.5 \times 0.8 = 0.4.
  • Pr(C&U) = 0.5 \times 0.2 = 0.1.
  • Pr(U) = Pr(S&U) + Pr(C&U) = 0.4 + 0.1 = 0.5.
  • Pr(S | U) = Pr(S&U) / Pr(U) = 0.4 / 0.5 = 0.8.
• Conclusion: Strong evidence that it's the Steroid team.

Two in a Row (with replacement)

• Scenario: Urns (from page 51). Pick an urn, draw two balls with replacement. Get Red, then Red again (R1&R2).
• Question: Pr(A | R1&R2)?
• Solution:
  • Pr(A&R1&R2) = Pr(A) * Pr(R1/A) * Pr(R2/A) (due to replacement) = 0.5 \times 0.8 \times 0.8 = 0.32.
  • Pr(B&R1&R2) = Pr(B) * Pr(R1/B) * Pr(R2/B) = 0.5 \times 0.4 \times 0.4 = 0.08.
  • Pr(R1&R2) = 0.32 + 0.08 = 0.4.
  • Pr(A | R1&R2) = Pr(A&R1&R2) / Pr(R1&R2) = 0.32 / 0.4 = 0.8.
• Learning from Experience: Pr(A | R) was 2/3; Pr(A | R1&R2) is 0.8. The second red ball increased the conditional probability of Urn A, showing how evidence strengthens belief.

The Gambler's Fallacy Once Again (revisited)

• Gambler thought Pr(red on 13th | 12 blacks) > 1/2$. But for independent trials, it remains$1/2.
• This highlights the core logical error in the fallacy: confusing the overall long-run distribution with the probability of individual independent events.

Two Weightlifters (without replacement)

• Scenario: 10 members per team. Test two randomly selected members.
• Assumptions: Steroid team (8 users, 2 non-users), Cleaner team (2 users, 8 non-users).
• Calculations:
  • Pr(U1&U2 | S) = (8/10$)$\times$($7/9$) =$28/45.
  • Pr(U1&U2 | C) = (2/10$)$\times$($1/9$) =$1/45.
  • Pr(S&U1&U2) = Pr(S) \times Pr(U1&U2 | S) = 0.5 \times 28/45 = 28/90.
  • Pr(C&U1&U2) = Pr(C) \times Pr(U1&U2 | C) = 0.5 \times 1/45 = 1/90.
  • Pr(U1&U2) = 28/90 + 1/90 = 29/90.
  • Pr(S | U1&U2) = Pr(S&U1&U2) / Pr(U1&U2) = (28/90) / (29/90) = 28/29 \approx 0.96.
• Conclusion: Two users testing positive provides very strong evidence (probability from 0.8$to$0.96) that it's the Steroid team.

6 The Basic Rules of Probability

Summary of Rules

• Rules apply to finite groups of propositions/events.
• Assumes basic deductive logic/set theory (logically equivalent propositions have same probability).

1. Normality

• 0 \le \text{Pr(A)} \le 1
• A measure normalized to be between 0$and$1.

2. Certainty

• Pr(certain proposition) = 1
• Pr(sure event) = 1$(often denoted Pr($\Omega$) =$1).

3. Additivity

• If A and B are mutually exclusive, then \text{Pr(AvB)} = \text{Pr(A)} + \text{Pr(B)}.

Overlap (Deduce from 1$-$3$)

• Pr(AvB) = Pr(A) + Pr(B) – Pr(A&B)
• Proof: AvB is logically equivalent to (A&B) v (A&~B) v (~A&B). These three components are mutually exclusive, so their probabilities can be added. Manipulating this using equivalences for A and B leads to the formula.

$5$. Conditional Probability (Definition)

• If Pr(B) > $0$, then $\text{Pr(A/B)} = \operatorname{Pr(A\&amp;B)} / \operatorname{Pr(B)}$.

$6$. Multiplication (Implied by 5$)

• If Pr(B) > 0$, then$\text{Pr(A\&B)} = \text{Pr(A/B)}\operatorname{Pr(B)}.

7$. Total Probability (Consequence of$5$)

• If 0 < \text{Pr(B)} < 1, then $\text{Pr(A)} = \text{Pr(B)}\operatorname{Pr(A/B)} + \text{Pr(~B)}\operatorname{Pr(A/~B)}$.
  • Useful for calculating Pr(A) when it depends on mutually exclusive circumstances (B or ~B).

Logical Consequence

• If B logically entails A, then $\text{Pr(B)} \le \text{Pr(A)}$.
  • This is shown because if B entails A, then B is equivalent to A&B. Thus Pr(A) = Pr(B) + Pr(A&~B), meaning Pr(A) is greater than or equal to Pr(B).

$8$. Statistical Independence (Definition)

• If 0 < \text{Pr(A)} and 0 < \text{Pr(B)}, A and B are statistically independent if and only if $\text{Pr(A/B)} = \text{Pr(A)}$.
  • Symmetry Proof: If Pr(A/B) = Pr(A), then Pr(B/A) = Pr(B).

Conditionalizing the Rules

• Rules (1)-(3) and (5) also hold in conditional form (e.g., replace Pr(A) with Pr(A/E)).
  • Normality (1C): $0 \le \text{Pr(A/E)} \le 1$.
  • Certainty (2C): Pr($[sure event]/E$) = $1$.
  • Additivity (3C): If A and B are mutually exclusive, then $\text{Pr((AvB)/E)} = \text{Pr(A/E)} + \text{Pr(B/E)}$.
  • Conditional Probability (5C): If Pr(E) > $0$ and Pr(B/E) > $0$, then $\text{Pr(A/(B\&amp;E))} = \operatorname{Pr((A\&amp;B)/E)} / \operatorname{Pr(B/E)}$.
• This highlights that some philosophers treat conditional probability as the primitive idea.

Multiple Independence

• Pairwise Independence: Defined by rule (8).
• Mutual Independence: A group of events (e.g., A, B, C) are mutually independent if all pairs are independent and $\text{Pr(A\&amp;B\&amp;C)} = \text{Pr(A)}\operatorname{Pr(B)}\operatorname{Pr(C)}$.

Venn Diagrams

• Origin: John Venn (1866) used them for logical arguments; also useful for probability.
• Representation: Circles represent classes/events; overlapping regions represent shared membership.
• Area as Probability: Shaded area represents probability, with the enclosing rectangle's area = $1$.
• Example (Musicians): $8$ musicians ($4$ singers only, $3$ whistlers only, $1$ singer & whistler).
  • Pr(singer selected) = $5/8$.
  • Pr(singer | whistler) = $1/4$ (1 singer&whistler out of $4$ total whistlers).
• Illustrating Basic Rules:
  • Normality: Circle areas within unit-area rectangle ($0 \le \text{Pr(A)} \le 1$).
  • Certainty: Event filling the entire rectangle has Pr = $1$.
  • Additivity: For mutually exclusive groups, combined area is sum of individual areas.
  • Overlap: Pr(AvB) = Pr(A) + Pr(B) – Pr(A&B).
  • Conditional Probability: Pr(A/B) = Area($\text{A\&amp;B}$) / Area(B).

Odd Question $2$: Pia's Profile (revisited)

• Scenario: Pia's detailed profile.
• Common Error (Tversky & Kahneman): Many people rank (f) (works in small bookstore, active feminist, takes yoga) as most probable.
• Logical Consequence Rule: $\text{Pr(A\&amp;B)} \le \text{Pr(B)}$. Thus, any conjunction is less than or equal to its conjuncts.
  • (f) logically entails (a), (c), (d), (e). So, Pr(a) > Pr(f), Pr(c) > Pr(f), etc.
  • Pr(a) $\ge$ Pr(d) $\ge$ Pr(e).
  • Pr(b) $\ge$ Pr(d) $\ge$ Pr(e).
  • Pr(a) $\ge$ Pr(f).
  • Pr(c) $\ge$ Pr(f).
• Possible Explanation for Error: People might be answering a "most useful/instructive/likely to be true" question rather than a strict probability ranking. Being informative often means providing specifics, even if less probable.

Axioms: Huygens (1657)

• First to publish axioms for probability, based on expected value/fair price.

Axioms: Kolmogorov (1933)

• Definitive axioms for probability theory, applying to infinite sets and calculus (measure theory).

7 Bayes' Rule

• A very useful consequence of the basic rules, key to "learning from experience."
• Examples from Chapter 5: Urns, shock absorbers, weightlifters all shared a common structure suitable for Bayes' Rule.
• Hypotheses (H): In general, $H_i$ represents competing hypotheses (e.g., Urn A vs. Urn B).
• Evidence (E): Event or observation (e.g., drawing a red ball).

Bayes' Rule (for Two Hypotheses)

• Let H and ${\sim}H$ be two mutually exclusive and exhaustive hypotheses.
• If Pr($\text{E}$) > $0$, then: $\operatorname{Pr(H/E)} = \frac{\operatorname{Pr(H)}\operatorname{Pr(E/H)}}{\operatorname{Pr(H)}\operatorname{Pr(E/H)} + \operatorname{Pr({\sim}H)}\operatorname{Pr(E/{\sim}H)}}$.

Proof of Bayes' Rule

• Derived directly from the definition of conditional probability and the rule of total probability.

Generalization of Bayes' Rule

• For $k$ mutually exclusive and jointly exhaustive hypotheses ($H<em>1, H</em>2, …, H<em>k$): $\operatorname{Pr(H</em>k/E)} = \frac{\operatorname{Pr(H<em>k)}\operatorname{Pr(E/H</em>k)}}{\sum<em>{i=1}^{k}[\operatorname{Pr(H</em>i)}\operatorname{Pr(E/H_i)}]}$ (where $\Sigma$ means sum).
• Bayes' Rule is a combination of basic rules, fundamental in some inductive logic theories.

Urns Example (revisited with Bayes' Rule)

• Pr(A) = Pr(B) = $0.5$. Pr(R/A) = $0.8$. Pr(R/B) = $0.4$.
• Question: Pr(A/R)?
• Solution: $(0.5 \times 0.8) / ((0.5 \times 0.8) + (0.5 \times 0.4)) = 0.4 / (0.4 + 0.2) = 0.4 / 0.6 = 2/3$.

Spiders Example

• Scenario: Bananas from Honduras (40%, 3% tarantulas) or Guatemala (60%, 6% tarantulas).
• Given: Tarantula found (T).
• Question: Pr(from Guatemala | Tarantula)? (Pr(G/T)?)
• Solution:
  • Pr(G) = $0.6$, Pr(H) = $0.4$.
  • Pr(T/G) = $0.06$, Pr(T/H) = $0.03$.
  • Pr(G/T) = $(0.6 \times 0.06) / ((0.6 \times 0.06) + (0.4 \times 0.03)) = 0.036 / (0.036 + 0.012) = 0.036 / 0.048 = 3/4$.

Taxicabs: Odd Question $5$ (revisited)

• Scenario: Green Cabs (85%), Blue Taxi (15%). Witness says blue. Witness $80$% accurate (Wb/B = $0.8$).
• Question: Pr(Blue | Witness says blue)? (Pr(B/Wb)?)
• Common Error (Tversky & Kahneman): People often think (a) $0.8$ or (b) > $0.5$ but < $0.8$. Very few pick (d) (more likely green).
• Solution:
  • Pr(G) = $0.85$, Pr(B) = $0.15$.
  • Pr(Wb/B) = $0.8$ (correctly identifies blue).
  • Pr(Wb/G) = $0.2$ (incorrectly identifies green as blue).
  • Pr(B/Wb) = Pr(B)Pr(Wb/B) / (Pr(B)Pr(Wb/B) + Pr(G)Pr(Wb/G)) = ($0.15 \times 0.8$) / (($0.15 \times 0.8$) + ($0.85 \times 0.2$))
    = $0.12 / (0.12 + 0.17) = 0.12 / 0.29 \approx 0.41$.
  • Pr(G/Wb) $\approx 1 - 0.41 = 0.59$.
• Conclusion: It is more likely that the sideswiper was green ($0.59$) than blue ($0.41$). So (d) is correct.

Base Rates

• Explanation of Error: People tend to ignore base rate information (most cabs are green). The witness's reliability (Pr(Wb/B)) is confused with the reliability of the statement (Pr(B/Wb)).
• Frequency Approach: Visualizing $100$ cabs (15 blue, 85 green) and applying witness accuracy (12 correctly blue, 17 incorrectly blue) makes the Bayesian answer clearer.

False Positives

• Medical Diagnosis Example: Test is $99$% accurate. If I have disease, test is YES $99$%. If I don't, test is NO $99$%. Test says YES.
• Disease Rarity: Base rate for disease = $1/10,000$.
• Confusion: Terrified by Pr(YES/I'm sick) (Idea $1$ reliability), but relieved by Pr(I'm sick/YES) (Idea $2$ reliability).
• Calculation: Pr($\sim$D | Y) = Pr($\sim$D)Pr(Y/$\sim$D) / (Pr($\sim$D)Pr(Y/$\sim$D) + Pr(D)Pr(Y/D))
  • $(9999/10000 \times 0.01) / ((9999/10000 \times 0.01) + (1/10000 \times 0.99)) = 9999 / (9999 + 99) \approx 0.99$ (probability of not having disease given YES).
• Conclusion: Even highly reliable tests can be misleading with very low base rates, yielding many false positives.

Strep Throat: Odd Question $6$ (revisited)

• Scenario: Physician suspects strep (Pr(S)=$0.9$). Test: $70$% YES if strep (Y/S=$0.7$), $90$% NO if no strep (N/$\sim$S=$0.9$). Results: YES, NO, YES, NO, YES.
• Assumptions: Test outcomes are independent.
• Likelihoods:
  • Pr(YNYNY | S) = $0.7 \times 0.3 \times 0.7 \times 0.3 \times 0.7 = 0.03087$.
  • Pr(YNYNY | $\sim$S) = $0.1 \times 0.9 \times 0.1 \times 0.9 \times 0.1 = 0.00081$.
• Using Bayes' Rule:
  • Pr(S | YNYNY) = ($0.9 \times 0.03087$) / (($0.9 \times 0.03087$) + ($0.1 \times 0.00081$)) $\approx 0.997$.
• Conclusion: Very much more likely than not that patient has strep throat (d).

"Sheer Ignorance" (Strep Throat with Pr(S)=$0.5$)

• If initial belief is $50-50$ (Pr(S)=$0.5$).
• Pr(S | YNYNY) = ($0.5 \times 0.03087$) / (($0.5 \times 0.03087$) + ($0.5 \times 0.00081$)) $\approx 0.974$.
• The test results are still powerful evidence, regardless of initial ignorance.

Rev. Thomas Bayes (1702-1761)

• English minister, interested in probability and induction.
• His posthumously published essay (1763) contained a solution to a sophisticated problem similar to current Bayes' Rule applications.
• The rule now known as Bayes' Rule is a simplification of his work.

8 Expected Value

Acts

• Definition: A decision to do something (or nothing).
• Consequences: Acts have consequences (desirable or not).
• Utility: The numeric cost or benefit of a consequence.
• Expected Value: Combines probabilities and utilities to evaluate acts. It is the sum of (probability $\times$ utility) for each consequence of an act.

Notation

• Acts: Bold capital letters (e.g., A).
• Consequences: Capital letters (e.g., C).
• Utility of C: U(C).
• Probability of C if A is taken: Pr(C/A).
• Expected Value of A: Exp(A).
• Formula (two consequences): Exp(A) = [Pr($\text{C}<em>1$/A)][U($\text{C}</em>1$)] + [Pr($\text{C}<em>2$/A)][U($\text{C}</em>2$)]. Simpler: Exp(A) = [Pr($\text{C}<em>1$)][U($\text{C}</em>1$)] + [Pr($\text{C}<em>2$)][U($\text{C}</em>2$)].
• Expected value is a weighted average of utilities.

A Free Ride (Lottery Ticket)

• Scenario: Aunt offers free lottery ticket. $100$ tickets, $90$ prize.
• Act: Accept ticket.
  • Consequence 1: Win ($90$ prize), Probability = $0.01$.
  • Consequence 2: Lose ($0$ prize), Probability = $0.99$.
• Exp(accepting) = ($0.01 \times 90$) + ($0.99 \times 0$) = $0.90$ or $90$¢.
• Exp(not accepting) = $0$.

Fair's Fair (Buying a Ticket)

• Scenario: Aunt sells ticket for $1.
• Exp(buying for $1) = ($0.01 \times (90-1)$) + ($0.99 \times -1)$) = $0.89 - 0.99 = -0.10$ or $-10$¢.
• Fair Price: The price at which the expected value of buying or selling is $0$. Here, $90$¢.
• Fair Price Arguments:
  1. Long Run: If played daily for years, average winning is $90$¢/game. If paid $1, lose $10¢/game on average.
  2. Buying All: If all $100$ tickets bought for $90$¢, total cost $90, total win $90. Breaks even, seems fair.

Generalizing Expected Value

• For $n$ mutually exclusive consequences: Exp(A) = $\sum \operatorname{Pr(C<em>i)}U(C</em>i)$.

Two Tickets (Lottery)

• Exp(accepting two free tickets) = $2 \times (0.01 \times 90) = 1.80$ or $1.80$.

Raffle Example

• Scenario: $100$ tickets. Prize 1: $90$. Prize 2: $9$.
• Exp(accepting one free ticket) = ($1/100 \times 90$) + ($1/100 \times 9$) = $0.90 + 0.09 = 0.99$ or $99$¢.
• Exp(buying one ticket for $1) = ($1/100 \times (90-1)$) + ($1/100 \times (9-1)$) + ($98/100 \times -1)$) = $0.89 + 0.08 - 0.98 = -0.01$ or $-1$¢.
• Fair price for this ticket is $99$¢.

Street Life (Illegal Vendor: Martin)

• Scenario: Sales $300/day, Merchandise cost $100 (Profit $200). Fine $100. Pr(ticket)=$0.4$.
• Act W: Go to work.
• Exp(W): Self-Employed (I):
  • $0.6 \times 200$ (no fine) + $0.4 \times (200-100)$ (fine) = $120 + 40 = 160$.
  • Alternative calculation: $200 - (0.4 \times 100) = 160$.
• Exp(W): Employed, Honest Boss (II):
  • Boss pays fines, charges Martin $50/day for cart.
  • Exp(W) = $200 - 50 = 150$.
  • Martin earns less but without risk here.

Expected Time of Travel

• Expected value applies to any quantifiable value, not just money.
• Scenario: Job interview in Ottawa. Train vs. Plane. Storm Pr=$0.2$.
  • Train: $30$ min to station, $10$ min departure, $5$ hours (no storm) / $7$ hours (storm) travel, $20$ min to interview.
  • Plane: $80$ min to airport, $1$ hour departure, $1$ hour (no storm) / $10$ hours grounded (storm) travel, $40$ min to interview.
• Exp(Train): Travel Time = $5$ or $7$ hours. Total base time (to station + departure + to interview) = $30 + 10 + 20 = 60$ min = $1$ hour.
  • Exp(Train) = $1 + (0.8 \times 5) + (0.2 \times 7) = 1 + 4 + 1.4 = 6.4$ hours ($6$ hrs $24$ min).
• Exp(Plane): Travel Time = $1$ or $10$ hours (grounded) + $1$ (flight). Total base time = $80 + 60 + 40 = 180$ min = $3$ hours.
  • Exp(Plane) = $3 + (0.8 \times 1) + (0.2 \times 11) = 3 + 0.8 + 2.2 = 6$ hours.
• Decision: Plane has shorter expected time. But other utilities matter (missing interview if grounded, nervousness during wait).

Roulette (Expected Value)

• North America (2 zeros): Exp($1$ bet on red) = ($18/38 \times 2$) - $1 \approx -5$¢.
• Europe (1 zero): Exp($1$ bet on red) = ($18/37 \times 2$) - $1 \approx -3$¢