Lesson 7: Understanding Uncertainty Study Notes - Detailed Study Guide on Probability, Expected Value, and the Law of Large Numbers

Introduction to Uncertainty and Probability

• Goal of Lesson 7: The primary objective is to understand uncertainty through the study of probability, expected value, and the Law of Large Numbers, including various real-world applications.
• Basic Probability Background:
  • Six-sided Dies: A standard die contains the numbers $1, 2, 3, 4, 5, \text{ and } 6$.
  • Standard Deck of Cards: Consists of $52$ total cards.
    • $26$ cards are black.
    • $26$ cards are red.
    • Each specific number or symbol (face card) appears $4$ times in the deck.
  • Utility of These Models: Dice and cards are frequently used in statistics because they serve as excellent simulation models for demonstrating fundamental probability rules.

Fundamental Rule 1: The Range and Definition of Probability

• The Bound Rule: Every probability must be a numerical value between $0$ and $1$.
  • Probabilities cannot be negative.
  • Probabilities cannot exceed $1$.
• Definition of Probability: It is the proportion of time that a specific event occurs.
• Calculation Formula:
  • $\text{Probability} = \frac{\text{Number of favorable events}}{\text{Total number of events}}$
• Approaches to Probability:
  • Classical Probability: This is based on pure logic and mathematics where all possible outcomes are known ahead of time and are equally likely. No physical data is required to calculate this.
    • Examples: The probability of flipping heads on a fair quarter is $1/2$. The probability of picking an Ace from a deck of $52$ cards is $4/52$.
  • Relative Frequency Probability: This requires real-world data to estimate the likelihood of an event because outcomes cannot simply be counted logicially.
    • Examples: Determining the probability of getting into a car accident while driving impaired, or finding the probability of a specific sample proportion (like $50\%$) among $50$ voters.
• Specific Examples and Exercises:
  • Picking a Red Ace: There are $2$ red aces in a deck of $52$.
    • $\text{Probability} = \frac{2}{52}$
  • Lottery Scenario: If $1,000$ students are randomly selected for free tickets out of $24,000$ who entered:
    • $\text{Probability} = \frac{1,000}{24,000} = \frac{1}{24} \text{ (approximately } 4\% \text{)}$
  • Legitimate Probability Values: Out of a list of options, only values between $0$ and $1$ (e.g., $0.5$) are legitimate.
  • Rolling a One on a Die: Since there is one occurrence of the number one out of six total equal outcomes:
    • $\text{Probability} = \frac{1}{6}$

Fundamental Rule 2: Mutually Exclusive Events and the Addition Rule

• Definition of Mutually Exclusive (Disjoint) Events: Two events are mutually exclusive if they cannot happen simultaneously.
• The Addition Rule for "Or": If outcomes are mutually exclusive, the probability that at least one occurs is the sum of their individual probabilities.
  • Formula: $P(A \text{ or } B) = P(A) + P(B)$
• The Danger of Double Counting: A common mistake is adding probabilities of events that are not mutually exclusive.
  • Example: If $70\%$) of students use Instagram and $60\%$) use TikTok, simply adding them results in $130\%$, which is impossible. This occurs because there is an overlap of students using both platforms.
• Specific Examples and Exercises:
  • Mutually Exclusive Outcomes A, B, and C: If $P(A) = 0.3$ and $P(B) = 0.5$, the probability of $P(A \text{ or } B) = 0.3 + 0.5 = 0.8$.
  • Teen TV Viewing Habits Study: If $17\%$) primarily watch TV on a tablet and $9\%$) primarily watch on an iPod Touch (assuming these are mutually exclusive primary categories), the probability of choosing a teen who uses either is $17\% + 9\% = 26\%$.
  • Adding Smartphone and Computer Usage: If $43\%$) watch on smartphones and $31\%$) watch on computers:
    • $\text{Probability} = 43\% + 31\% = 74\%$
  • Rolling Die Values: The probability of rolling a value less than $3$ (which is rolling a $1$ or a $2$):
    • $P(1) + P(2) = \frac{1}{6} + \frac{1}{6} = \frac{1}{3}$

Fundamental Rule 3: The Rule of Complements

• Definition of Complements: The chance of something occurring is $1$ minus the chance of the opposite thing occurring.
• Formula: $P(A^C) = 1 - P(A)$
• Practical Use Cases: Complements are helpful when calculating the target event is complex, but calculating the non-event is simple.
  • Drawing Cards: To find the probability of getting at least one Ace when drawing five cards, it is easier to calculate $1 - P(\text{no Aces})$ rather than summing the probabilities of exactly $1, 2, 3, 4, \text{ and } 5$ Aces.
  • Insurance Risk: It is easier to compute the probability that a $30$-year-old survives a year than to list and sum every possible cause of death.
  • Quality Control: It is easier to calculate total working parts than to compute probabilities for the many ways at least one part could fail.
• Specific Examples and Exercises:
  • Complement of C: If $P(C) = 0.2$, then the complement of $C$ (denoted as $C^C$) is $1 - 0.2 = 0.8$.
  • TV Study (Not Smartphone): If the probability of using a smartphone is $43\%$, the probability of not using a smartphone is $1 - 0.43 = 0.57 \text{ (or } 57\% \text{)}$.

Fundamental Rule 4: Independence and the Multiplication Rule

• Definition of Independence: Two events are independent if the knowledge that one event has occurred does not change the probability of the second event occurring. The process has "no memory."
• The Multiplication Rule for "And": When calculating the probability of two independent events both happening (indicated by the word "and"), you multiply their chances.
  • Formula: $P(A \text{ and } B) = P(A) \times P(B)$
• Independence vs. Human Intuition: Humans naturally look for patterns and assume things occurring together are related.
  • Gambler's Fallacy: After a roulette wheel lands on black six times, players feel red is "due." However, the wheel has no memory; each spin is independent.
  • Randomness Imagery: In a coin flip sequence, a random-looking pattern (Heads-Tails-Heads-Tails-Tails-Heads) feels more likely to humans than a repeating pattern (Heads-Heads-Heads-Heads-Heads-Heads), but for a fair coin, both sequences have a probability of $(\frac{1}{2})^6 = \frac{1}{64}$.
• Independence vs. Mutual Exclusivity:
  • Mutually Exclusive: Relates to possibility. Events cannot co-occur. They are highly dependent because if A happens, B is impossible ($P(A \text{ and } B) = 0$).
  • Independent: Relates to influence. One event does not affect the odds of the other.
• Testing for Independence:
  • Case Study (Pets): Consider $38\%$) of households own a dog ($P(D) = 0.38$), $25\%$) own a cat ($P(C) = 0.25$), and $17\%$) own both ($P(D \text{ and } C) = 0.17$).
  • To test for independence: $0.38 \times 0.25 = 0.095$.
  • Since $0.095 \neq 0.17$, owning a dog and owning a cat are not independent.
  • Conditional Change: Among cat owners, the chance of owning a dog jumps from $38\%$) to $17/25 = 68\%$, proving dependence.

Fundamental Rule 5: Subsets and Probability Constraints

• The Subset Rule: If the ways in which event B can happen are a subset of the ways event A can happen, then event B cannot have a higher probability than event A.
  • Formula: $P(B) \times \text{is a subset of } P(A) \rightarrow P(B) \times \text{must be } \times \text{less than or equal to } \times P(A)$
• The Conjunction Fallacy: A more detailed story often feels more representative of reality to humans, but it is mathematically less probable.
  • Example (James): James is a $45$-year-old logic puzzle lover. Is he more likely to be (A) an accountant or (B) an accountant and a soccer coach? Option A is more probable because B is a more restrictive subset of A.
  • Example (Marriage): The probability of getting married and having children must be less than or equal to the probability of just getting married.
• Two-Way Table Application:
  • World Campus Data (Fall 2019): Total students = $14,687$. World Campus students from PA = $6,010$.
  • $\text{Probability of picking a PA World Campus student} = \frac{6,010}{14,687}$.
  • The probability of picking a subset (PA World Campus undergraduates) is strictly less than the probability of picking all PA World Campus students.

Expected Value ($EV$)

• Definition: Expected value is a way of using probability to determine future long-term outcomes. It is the average value one would expect from an experiment repeated many times.
• Calculation Formula: The sum of each outcome multiplied by its respective probability.
  • $EV = \text{Sum of } (\text{Outcome}_i \times \text{Probability}_i)$
• Risk Analysis in Gains and Losses:
  • Gains: Humans typically prefer a sure gain over a riskier prospect, even if the $EV$ is higher on the riskier one.
    • Choice A: $\$240$ Guaranteed.
    • Choice B: $25\%$) chance to win $\$1,000$ ($EV = \$250$).
    • Most people choose A.
  • Losses: Humans are generally willing to take risks to avoid a certain loss.
    • Choice C: Sure loss of $\$740$.
    • Choice D: $75\%$) chance to lose $\$1,000$ ($EV = -\$750$).
    • Most people choose D.
• Detailed Examples:
  • Roulette Wheel: An American wheel has $38$ total slots ($18$ red, $20$ black/green).
  • Betting $\$1$ on red:
    • $EV = (\$1 \times \frac{18}{38}) + (-\$1 \times \frac{20}{38}) = -\$0.0526$
    • In the long run, the house makes approximately $5 \text{ cents}$ for every dollar bet.
  • Illegal Parking Tickets:
    • $80\%$) chance of $\$0$ ticket.
    • $15\%$) chance of $\$35$ ticket.
    • $5\%$) chance of $\$75$ ticket.
    • $EV = (0 \times 0.80) + (35 \times 0.15) + (75 \times 0.05) = 0 + 5.25 + 3.75 = \$9.00$
    • The expected average cost per time parked illegally is $\$9$.

Law of Large Numbers (LLN)

• Core Principle: As the number of independent trials increases, the experimental results (proportions and averages) will stabilize and settle closer to the true population value.
• Miles Per Gallon (MPG) Metaphor: After resetting a trip computer, the MPG is unstable for the first few miles. However, after $300 \text{ miles}$, it barely changes because it has settled into the true average for the car.
• Stable vs. Unstable Results:
  • Stable: Proportions or averages tend to get closer to the true value as trials increase.
  • Unstable: The raw sum or count of events tends to vary further from the true value as trials increase.
• Trials and Outcomes Comparison:
  • Small Number of Trials: Higher chance of results being exactly equal to the expected number (e.g., exactly $5$ heads in $10$ flips). Local results have a larger range of proportions/averages.
  • Large Number of Trials: Lower chance of hitting a count exactly (e.g., exactly $500$ heads in $1,000$ flips), but the proportion will have a very narrow range around the true mean.
• Simulation vs. Practical Probabilities:
  • Gas Station Example: If $60\%$) of customers fill their tanks:
    • Getting between $58\% \text{ and } 62\%$) is more likely with $1,000$ customers than $100$.
    • Getting exactly $60$ out of $100$ is more likely than exactly $600$ out of $1,000$.
  • Hospital Example: If $48\%$) of babies are girls, a majority (greater than $50\%$) is more likely to occur in a smaller sample of $20$ babies than in a larger sample of $200$, because the larger sample pulls the proportion closer to the true $48\%$.
  • Easy Pass Example: If $40\%$) use Easy Pass, a range of $35\% \text{ to } 45\%$) is more likely for $800$ cars than for $100$ cars.
  • Convenience Store Spending: If the average spent is $\$20$, the average of the next $283$ customers is more likely to stay between $\$16 \text{ and } \$24$ than the average of the next $112$ customers due to lower variability in higher trials.
  • Independence Test: If $10$ customers in a row spend less than $\$20$, the chance the next customer spends more or less remains equal (assuming a median of $\$20$) because trials are independent.

Summary Takeaways

• Relative Frequency: Probability represents long-run proportions.
• Expected Value: Outcome times probability represents long-run expected return.
• Law of Large Numbers: Independent trials lead to stable averages/proportions that close in on true population values.
• Independence vs. Dependence: Knowledge of one event either has no impact (independence) or changes the odds (dependence) of another.
• Mutually Exclusive: Events that cannot happen at the same time.