Introduction to Probability, Sample Spaces, Graphical Tools, and Complex Events

Fundamental Probability Concepts and Sample Spaces

• Experiment Defined:

  • An experiment is any process or procedure that generates uncertain outcomes.
  • If an event or outcome has a $100\%$ likelihood of occurrence ($P = 1$), it is strictly not classified as an experiment because it lacks outcome uncertainty.
  • Medical Treatment Example: Administering a specific treatment or medication to a patient to evaluate its efficacy is an experiment. Depending on various factors, medications may work or fail. The basic observed outcomes, such as recovery or non-recovery, represent the outcomes of the experiment.

• Sample Points:

  • A sample point is the most fundamental, atomic, and indivisible outcome of an experiment.
  • Coin Flip Example: Flipping a single coin yields exactly two basic sample points: Head ($H$) or Tail ($T$).
  • Die Roll Example: Rolling a standard six-sided die yields six distinct sample points: $1$, $2$, $3$, $4$, $5$, and $6$

• Sample Space:

  • The sample space is the complete set or collection containing all possible basic sample points of an experiment.
  • Notation: A sample space is formally written by enclosing all distinct sample points inside brackets: $\mathbf{S} = \{1, 2, 3, 4, 5, 6\}$.
  • Each unique sample point is listed only once within the sample space brackets.

• Likelihood (Probability) Bounding and Rules:

  • Likelihoods are real numbers strictly bounded between $0$ and $1$ inclusive ($0 \le P \le 1$).
  • $P = 0$: Represents an outcome that is guaranteed not to happen.
  • $P = 1$ (or $100\%$): Represents an absolute certainty where nothing else can possibly occur.
  • $0 < P < 1$: Represents the probability range for standard uncertain real-world events. For instance, a $40\%$ chance of rain tomorrow corresponds to a likelihood of $0.40$, which satisfies $0 < 0.40 < 1$
  • Summation Axiom: The sum of the individual likelihoods of all sample points in a sample space must equal exactly $1$ (or $100\%$).
    • For a coin: $P(\text{Head}) + P(\text{Tail}) = 1$
    • For a six-sided die: $P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1$

• Fair vs. Unfair Models:

  • Fair Coin: Both outcomes are equally likely, yielding a $50\%$ chance ($0.50$) for Head and a $50\%$ chance ($0.50$) for Tail.
  • Unfair Coin: Probabilities are asymmetric across sample points (e.g., $80\%$ and $20\%$, or $35\%$ and $65\%$).
  • Fair Die: Every outcome has an identical probability of \n\frac{1}{6}  because the six identical probabilities must sum to $1$ ($6 \times \frac{1}{6} = 1$).
  • Unfair Die: Probabilities assigned to sample points deviate from \n\frac{1}{6} , though their sum still equals $1$

Event Definition and Scope

• Event Concept and Notation:

  • An event is created by grouping specific sample points together based on defined rules or criteria.
  • Notation: Standard convention utilizes a single capital letter (e.g., $A$, $B$, $C$) or capital $E$ to denote an event.
  • Complexity: Events range from simple events (containing a single sample point or a few sample points) to complex compound events.
  • Scope Limit: Probability analyses in this content focus on single events and relationships between two events, explicitly avoiding three-event pools.

• Die Roll Event Definitions:

  • Sample Space: $\mathbf{S} = \{1, 2, 3, 4, 5, 6\}$
  • Event $A$ (Even Numbers): Contains sample points $\mathbf{A} = \{2, 4, 6\}$
  • Event $B$ (Odd Numbers): Contains sample points $\mathbf{B} = \{1, 3, 5\}$
  • Event $C$ (Numbers Greater Than $3$): Contains sample points $\mathbf{C} = \{4, 5, 6\}$

Graphical Tools for Probability Analysis

• Tree Diagrams:

  • Function: A step-by-step graphical tool used to break down sequential multi-stage processes and trace all outcomes without jumping prematurely to final conclusions.
  • Application - Two Fair Coins:
    • Stage 1 (First Coin): Branches into Head ($H$) or Tail ($T$) (2 branches).
    • Stage 2 (Second Coin): Each Stage 1 outcome branches again into Head ($H$) or Tail ($T$).
    • Total Final Branches: $2 \times 2 = 2^2 = 4$ total outcomes.
    • Branch Outcomes: Head-Head ($HH$), Head-Tail ($HT$), Tail-Head ($TH$), and Tail-Tail ($TT$).
    • Probability Calculation Along Path: Since each coin flip has a likelihood of \n\frac{1}{2} , multiplying probabilities along the path yields the likelihood of a specific final sample point:             \n\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\n

• Venn Diagrams:

  • Function: A visual diagram used to analyze spatial relationships between events, such as event overlap, non-overlap, and relative likelihoods.
  • Structure: Encloses sample points or events within closed shapes ("bubbles"). The relative physical size of a bubble visually corresponds to the likelihood of that event occurring.
  • Interactions in Die Roll Example:
    • Event $A$ (Even: $\mathbf{A} = \{2, 4, 6\}$) and Event $B$ (Odd: $\mathbf{B} = \{1, 3, 5\}$): Do not overlap at all because even and odd numbers cannot happen simultaneously.
    • Event $A$ (Even) and Event $C$ ($> 3$: $\mathbf{C} = \{4, 5, 6\}$): Overlap at sample points $4$ and $6$ (numbers that are both even and greater than $3$).
    • Event $B$ (Odd) and Event $C$ ($> 3$): Overlap at sample point $5$ (a number that is both odd and greater than $3$).

Complex Events: Union and Intersection

• Mathematical Translation of English Keywords:

  • Translating word problems into formal mathematical notation requires strict attention to precise phrasing. A change of a single letter or word completely alters the operation and result.

• Union ($A \cup B$):

  • Keyword: "or".
  • Definition: Includes outcomes that belong to Event $A$, Event $B$, or both.
  • Visual: Represented in a Venn diagram by the total area covered by both bubbles combined, including their overlap.

• Intersection ($A \cap B$):

  • Keyword: "and".
  • Definition: Requires both Event $A$ and Event $B$ to occur simultaneously.
  • Visual: Represented in a Venn diagram by only the shared overlapping region between the event bubbles.

• Case Example: Toxic Chemical Incidents in Taiwan:

  • Context: Categorization of toxic chemical incident occurrences in Taiwan based on plant facility type/location.
  • Total Probability: The sum of all location likelihoods equals $100\%$ ($1$).
  • Problem Scenario: Determine the probability that a toxic chemical incident occurs in either a chemical plant or a non-chemical plant.
  • Analysis: The keyword "or" signifies a Union operation ($A \cup B$).
  • Calculation: Because chemical plant incidents and non-chemical plant incidents do not overlap (they are mutually exclusive classifications), the total union probability is calculated by adding their individual likelihoods directly:         \n P(A \cup B) = P(A) + P(B) \n

Questions and Discussion

• Question: If something has a $100\%$ likelihood of happening, is it considered an experiment?

  • Answer: No. An experiment requires uncertain outcomes. If an event has a $100\%$ probability of occurring, there is no uncertainty, so it does not meet the definition of an experiment.

• Question: Are the lecture slides available online?

  • Answer: Yes, the slides are posted online. If specific slides are hidden, it is because they contain annotated work from last year that was intentionally hidden to avoid overwhelming students.