Comprehensive Study Notes on Probability Theory, Empirical Models, and Addition Rules

Sample Spaces and Event Set Notation

• Definition of Sample Space ($S$):

  • The sample space of an experiment or random phenomenon is the set of all possible outcomes.
  • Mathematically expressed using set notation as $S = \{E_1, E_2, \dots, E_n\}$.

• Ordered Pairs and Outcomes in Multi-Child Families:

  • When analyzing the gender of children in a family, birth order (older versus younger) dictates that outcomes are ordered pairs.
  • For a two-child family, the sample space $S$ consists of $4$ distinct ordered pair outcomes:         $S = \{(B, B), (B, G), (G, B), (G, G)\}$
  • $B$ represents a Boy, and $G$ represents a Girl.
  • The ordered pair $(B, G)$ denotes that the firstborn child is a boy and the secondborn child is a girl, which is distinct from $(G, B)$ where the firstborn is a girl and the secondborn is a boy.
  • Expanding the analysis to three children creates a larger tree structure with branching outcomes yielding $2^3 = 8$ total outcomes (e.g., $(B, B, B)$, $(B, B, G)$, etc.).

• Mathematical Definition of an Event:

  • An event is any subset of the sample space $S$.
  • An event can consist of a single outcome, multiple outcomes, all outcomes in $S$, or none of the outcomes (the empty set).
  • Example Event: Defining the event "have exactly one boy" in a two-child family.
    • Written in set notation, this event $E$ contains $2$ ordered pairs:             $E = \{(B, G), (G, B)\}$

Fundamental Rules and Models of Probability

• Probability Range Property (Axiom 1):

  • The probability of any event $A$, denoted $P(A)$, is a real number between $0$ and $1$ inclusive:         $0 \le P(A) \le 1$
  • Expressed as a percentage, probability ranges strictly from $0\%$ to $100\%$.
  • Percentage conversion is defined by the division ratio over $100$:         $100\% = \frac{100}{100} = 1$

• Sum of Probabilities Property (Axiom 2):

  • The sum of the probabilities of all mutually exclusive simple outcomes in a complete sample space $S = \{E_1, E_2, \dots, E_n\}$ must equal $1$ (or $100\%$):         $\sum_{i=1}^{n} P(E_i) = 1$
  • If a list of probabilities for all listed outcomes sums to $97\%$ ($0.97$), the outcome list is incomplete, indicating that $3\%$ ($0.03$) of the sample space is missing.

• Probability Models:

  • A probability model consists of a sample space and a probability assignment to each outcome.
  • To be mathematically valid, a probability model must satisfy both essential conditions:
    1. Every individual probability $P(E_i)$ satisfies $0 \le P(E_i) \le 1$
    2. The sum of all outcome probabilities satisfies $\sum P(E_i) = 1$

Empirical vs. Classical Approaches to Probability

• Classical (Theoretical) Approach:

  • Used when all outcomes of an experiment are equally likely.
  • Does not require conducting physical experiments to compute likelihoods.
  • Formula for classical probability of event $E$:         $P(E) = \frac{\text{Number of outcomes in } E}{\text{Total number of outcomes in } S}$
  • Example: Rolling a fair six-sided die where each side has equal likelihood:         $P(4) = \frac{1}{6} \approx 0.1667 \,\text{or}\, 16.67\%$

• Empirical (Relative Frequency) Approach:

  • Used when outcomes are not equally likely or when theoretical probabilities cannot be derived analytically.
  • Based on direct observation and repeated experimentation.
  • Relative frequency is computed as the count of observed occurrences $f$ divided by total trials $n$:         $\text{Relative Frequency} = \frac{f}{n}$
  • Law of Large Numbers: As the number of times an experiment is repeated ($n$) increases, the observed empirical relative frequency approaches the true long-term probability:         $\lim_{n \to \infty} \frac{f}{n} = P(E)$
  • Probability can be expressed in three mathematically equivalent formats: fractions/ratios, decimals, and percentages.
  • Short-Term vs. Long-Term Predictability: Short-term outcomes of a random process are uncertain and impossible to predict individually; long-term relative frequencies stabilize and display predictable regularities.

Practical Application: The Pass the Pigs Experiment

• Experiment Overview:

  • In the game Pass the Pigs, asymmetrical pig-shaped game pieces are rolled similarly to dice.
  • A single pig can land in six distinct physical positions:
    1. Razorback / Raised back (landing on its back)
    2. Side with dot
    3. Side without dot
    4. Trotter (standing upright on its feet)
    5. Snouter (balanced on its snout and front feet)
    6. Leaning Jowler (leaning on its snout and jaw)

• Likelihood and Physical Mechanics:

  • Lowest probability outcome: Leaning Jowler is the hardest position to obtain physically.
  • Highest probability outcomes: Side positions (Side with dot and Side without dot) are the easiest and most frequent outcomes.

• Classroom Data Analysis:

  • An experiment was performed by a class of $52$ students.
  • Total tosses performed and recorded: $3,939$ pig rolls.
  • Specific outcome counts recorded in data tables included $32$ occurrences for a low-probability landing position.
  • The empirical relative frequency calculated for landing on "Side with dot" was $0.329$ ($32.9\%$, approximately $33\%$).

• Long-Term Interpretation of $0.329$ ($33\%$) Probability:

  • If the pig is tossed $1,000$ times, it is expected to land on "Side with dot" at least $329$ times.
  • If rolled $1,000,000$ times in the long term, "Side with dot" will appear in approximately $33\%$ of the total rolls.

Testing Die Fairness via Empirical Observation

• Fair Die Expectations:

  • For a fair, balanced six-sided die, the theoretical probability of landing on any face is $\frac{1}{6} \approx 0.1667$ ($16.67\%$).
  • If a fair die is rolled $100$ times, each individual number ($1$ through $6$) is expected to appear roughly $16$ to $17$ times:         $\text{Expected count} = 100 \times \frac{1}{6} \approx 16.67\,\text{times}$
  • Empirical counts between $15$ and $18$ (e.g., $17$, $16$, $15$, $18$) across $100$ rolls confirm fair behavior.

• Weighted (Unfair) Die Mechanics:

  • A weighted die has unequal weight distribution across its geometry.
  • Physical tendency: The heavier side tends to land facing down due to gravity, forcing the opposite side to land facing up with a significantly higher probability.
  • Empirical detection: If a die is tested over $100$ trials and the number $6$ appears only $2$ times (far lower than the expected $16.67$ times), the die is empirically proven to be loaded/unfair.

Compound Events and Probability Rules

• Classification of Compound Events:

  • Union ($A \text{ or } B$ / $A \cup B$): Event containing outcomes belonging to $A$, to $B$, or to both. Analyzed using the Addition Rule.
  • Intersection ($A \text{ and } B$ / $A \cap B$): Event containing outcomes belonging strictly to both $A$ and $B$ simultaneously. Analyzed using the Multiplication Rule.
  • Conditional Probability ($A \mid B$): Probability of event $A$ occurring given that event $B$ has already occurred.

• Venn Diagram Representations:

  • Sample space $S$ is visualized as a surrounding rectangle (the universe).
  • Outcomes are individual points distributed within $S$
  • Events $A$ and $B$ are represented as circular regions containing specific outcome points.

• Addition Rule for Disjoint (Mutually Exclusive) Events:

  • Events $A$ and $B$ are disjoint if they share no common outcomes ($A \cap B = \emptyset$).
  • To find $P(A \text{ or } B)$, count outcomes in $A$ ($N(A)$) plus outcomes in $B$ ($N(B)$) divided by total outcomes $N(S)$:         $P(A \text{ or } B) = \frac{N(A) + N(B)}{N(S)} = \frac{N(A)}{N(S)} + \frac{N(B)}{N(S)} = P(A) + P(B)$

• General Addition Rule for Non-Disjoint Events:

  • When events $A$ and $B$ overlap ($A \cap B \neq \emptyset$), outcomes in the intersection belong to both events.
  • Example Scenario: Rolling a standard six-sided die with sample space $S = \{1, 2, 3, 4, 5, 6\}$.
    • Event $A$ (Even number) = {2, 4, 6} ($N(A) = 3$)
    • Event $B$ (Number $\le 4$) = {1, 2, 3, 4} ($N(B) = 4$)
    • Intersection $A \cap B$ = {2, 4} ($N(A \cap B) = 2$)
  • Double-Counting Problem: Simply adding $N(A) + N(B)$ counts outcomes in the intersection ($A \cap B$) twice.
  • Correction Step: Subtract the count of the intersection once to eliminate double-counting:         $N(A \text{ or } B) = N(A) + N(B) - N(A \cap B)$
  • Dividing through by total outcomes $N(S)$ yields the General Addition Rule Formula:         $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Questions & Audience Discussion

• Interpretation of Event Criteria:

  • Question: In defining the event "have one boy," does it mean "at least one boy" or "strictly/only one boy"?
  • Response: It refers strictly to "only one boy." In a two-child sample space $S = \{(B, B), (B, G), (G, B), (G, G)\}$, the event contains exactly $2$ ordered pairs: \Rule{0pt}{0pt}\{(B, G), (G, B)\}.

• Pass the Pigs Relative Probability Evaluation:

  • Question: Which pig position has the absolute lowest chance of occurring during a toss?
  • Response: The