Relative Frequency, Discrete Probability Distributions, Contingency Tables, and Conditional Probability

Fundamentals of Relative Frequency and Coin-Tossing Experiments

  • Sum of Relative Frequencies Axiom:

    • The sum of all relative frequencies for all possible outcomes in a sample space must always equal 11.

    • Fundamental formula:     ∑Relative Frequency=1\sum \text{Relative Frequency} = 1

    • When a relative frequency value is missing from a distribution table, it is calculated by subtracting the sum of all known relative frequencies from 11:     Missing Relative Frequency=1−∑Known Relative Frequencies\text{Missing Relative Frequency} = 1 - \sum \text{Known Relative Frequencies}

  • Coin-Tossing Experiment Analysis:

    • Event: Second coin lands heads up:

    • The outcome of the first coin is irrelevant when evaluating an event that depends solely on the second coin.

    • To find the relative frequency of this event, sum the relative frequencies of all outcomes where the second coin displays heads.

    • Calculation:       Relative Frequency=0.275+0.30=0.575\text{Relative Frequency} = 0.275 + 0.30 = 0.575

    • Event: Tails comes up at least once:

    • "At least once" signifies that tails appears either 11 time or 22 times across the coin flips.

    • Favorable outcomes satisfying this condition: {Heads-Tails,Tails-Heads,Tails-Tails}\{\text{Heads-Tails}, \text{Tails-Heads}, \text{Tails-Tails}\}.

    • Direct Method: Sum the relative frequencies of all three outcomes containing at least one tail.

    • Complement Method (Short Way): Subtract the relative frequency of the single unincluded outcome (zero tails, which is Heads-Heads\text{Heads-Heads}) from 11:       Relative Frequency=1−0.275=0.725\text{Relative Frequency} = 1 - 0.275 = 0.725

    • Expressed as a percentage, this represents a 72.5%72.5\% chance (approximately 73%73\%).

    • Efficiency Principle: When an event encompasses the majority of outcomes in a sample space, calculating 1−unwanted outcome1 - \text{unwanted outcome} is computationally simpler than summing all individual favorable outcomes.

Discrete Probability Distributions and Event Complements

  • Missing Relative Frequency in Outcome Tables:

    • Consider a discrete distribution with outcomes {1,2,3,4,5}\{1, 2, 3, 4, 5\}.

    • Given known relative frequencies such as 0.10.1, 0.30.3, and 0.40.4, sum the known relative frequencies:     0.1+0.1+0.1+0.3+0.4=1.00.1 + 0.1 + 0.1 + 0.3 + 0.4 = 1.0

    • If known relative frequencies sum to 0.90.9, the missing relative frequency is calculated as:     Missing Value=1−0.9=0.1\text{Missing Value} = 1 - 0.9 = 0.1

  • Relative Frequency of Compound Events:

    • For an event containing outcomes 22, 33, or 44 (denoted as event {2,3,4}\{2, 3, 4\}):

    • The individual outcomes are mutually exclusive (an outcome of 22 cannot simultaneously be 33 or 44).

    • The relative frequency of the compound event is the sum of their individual relative frequencies:       Relative Frequency=0.1+0.1+0.3=0.5\text{Relative Frequency} = 0.1 + 0.1 + 0.3 = 0.5

  • Event Complement Notation and Calculation:

    • Definition: The complement of an event EE, denoted by E′E', consists of all outcomes in the sample space that are not included in event E$.\n * Given event Edefinedascontainingoutcomesdefined as containing outcomes2andand4((E = {2, 4}):\n * Set representation of the complement: E' = {1, 3, 5}\n * Calculating the relative frequency of E':\n * Sum the relative frequencies corresponding to outcomes 1,,3,and, and5 directly from the distribution table.\n * Calculation:\n      \text{Relative Frequency}(E') = 0.6\n\n\n# Contingency Tables and Marginal vs. Joint Relative Frequencies\n\n* **Structure of a Two-Variable Author Survey**:\n * Author status is categorized into two categories: Successful (S)andUnsuccessful() and Unsuccessful (U).\n * Author experience is categorized into two categories: New (N)andEstablished() and Established (E).\n * Total sample size: N_{\text{total}} = 100\n\n* **Contingency Table Data Matrix**:\n * Successful and New (S \cap N):):5\n * Unsuccessful and New (U \cap N):):15\n * Successful and Established (S \cap E):):25\n * Unsuccessful and Established (U \cap E):):55\n * **Marginal Totals**:\n * Total New Authors (N):):5 + 15 = 20\n * Total Established Authors (E):):25 + 55 = 80\n * Total Successful Authors (S):):5 + 25 = 30\n * Total Unsuccessful Authors (U):):15 + 55 = 70\n * Total Overall Sample: 30 + 70 = 20 + 80 = 100\n\n* **Definitions and Mathematical Definitions**:\n * **Relative Frequency Formula**:\n    \text{Relative Frequency} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Sample Size}}\n * **Intersections (\cap)**: Represent overlapping conditions where two attributes occur simultaneously (e.g., an author who is both successful and new).\n * **Marginal Totals**: Row or column totals representing the absolute total frequency of a single attribute, regardless of the other variable category.\n\n* **Calculations of Standard Relative Frequencies**:\n * **Part A: Author is established and successful (E \cap S)**:\n * Count at the intersection of Established and Successful = 25\n * Calculation:\n      \text{Relative Frequency} = \frac{25}{100} = 0.25\n * **Part B: Author is new (N)**:\n * Total marginal count of new authors = 20\n * Calculation:\n      \text{Relative Frequency} = \frac{20}{100} = \frac{1}{5} = 0.20\n * **Part C: Author is unsuccessful (U)**:\n * Total marginal count of unsuccessful authors = 70\n * Calculation:\n      \text{Relative Frequency} = \frac{70}{100} = 0.70\n\n\n# Conditional Probability and Sample Space Reduction\n\n* **Concept of Conditional Probability**:\n * Conditional probability evaluates the likelihood of an event given that another condition is already known to be true.\n * Knowing that a condition is met restricts the total sample space from the overall sample size (100) to the specific marginal total of the known condition.\n\n* **Step-by-Step Conditional Calculations**:\n * **Part D: Given a successful author, find the relative frequency that the author is established (E \mid S)**:\n * Known condition: The author is successful (S).\n * Reduced sample space (denominator): Total successful authors = 30\n * Favorable outcomes within this sub-population (numerator): Successful authors who are established = 25\n * Calculation:\n      \text{Conditional Relative Frequency} = \frac{25}{30}\n * **Part E: Given an established author, find the relative frequency that the author is successful (S \mid E)**:\n * Known condition: The author is established (E).\n * Reduced sample space (denominator): Total established authors = 80\n * Favorable outcomes within this sub-population (numerator): Established authors who are successful = 25\n * Calculation:\n      \text{Conditional Relative Frequency} = \frac{25}{80}\n\n* **Order Sensitivity in Conditional Statements**:\n * The order of conditions significantly alters the calculated value:\n    \frac{25}{30} \neq \frac{25}{80}\n * While both calculations retain the same joint count in the numerator (25),thedenominatorchangesbasedonwhichsub−populationisgiven(), the denominator changes based on which sub-population is given (30forgivensuccessfulvs.for given successful vs.80$$ for given established).