Chapter 4: Introduction to Probability

• Probability quantifies uncertainty and underpins econometrics.

• It serves as a foundational topic for the course, indicating its importance for succeeding content throughout the semester.

4.1 Random Experiments, Counting Rules, and Assigning Probabilities

Random Experiments

• Definition: A random experiment yields outcomes that may vary, even under identical conditions.

• Probability Scale: Values range from 0 (unlikely) to 1 (certain).

• Sample Space: The complete set of possible outcomes from an experiment.

  • Examples:

    • Coin Toss: {Heads, Tails}

    • Die Roll: {1, 2, 3, 4, 5, 6}

    • Basketball Shot: {Make, Miss}

    • Lottery: [0, Jackpot]

    • Race: {1st, 2nd, 3rd, ... , Last}

Counting Outcomes

• Total outcomes for a sequence of steps:

  • If each step has different outcome possibilities: Total Outcomes = n1 * n2 * ... * nk

  • Example: For movie options at ACM Theaters: 3 movies x 3 snacks = 9 outcomes.

Assigning Probabilities

• Types of Probability Assignment:

  • Classical Method: Based on equally likely outcomes (e.g., rolling a die).

  • Relative Frequency Method: Based on historical data; probabilities should sum to 1.

  • Subjective Method: Based on personal judgment and expertise.

Example Calculation (Die Roll)

• For a 6-sided die: Each outcome has a probability of 1/6.

4.2 Events and Their Probabilities

• Event Definition: A collection of one or more sample points.

• The probability of an event equals the sum of the probabilities of its sample points.

  • Example: Probability of Collins Mining being profitable involves adding the probabilities of combinations of revenues and costs.

4.3 Basic Relationships of Probability

Key Concepts

• Complement of an Event: All sample points not in the event A, denoted by Ac.

• Union of Two Events: All sample points in A or B, denoted by A ∪ B.

• Intersection of Two Events: Sample points in both A and B, denoted by A ∩ B.

• Addition Law: For events A & B, the formula is:

  • P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

• Mutually Exclusive: Events that cannot occur simultaneously.

Addition Law in Practice

• Calculate probabilities in scenarios like getting a flat tire and overheating the car.

4.4 Conditional Probability

• Definition: The probability of event A given that B has occurred, denoted as P(A|B).

• Calculation Formula: P(A|B) = P(A ∩ B) / P(B)

• Example: Calculating likelihood of rain given that it’s cloudy:

  • Use known probabilities to compute.

Joint Probability Table

• Concept: Displays joint and marginal probabilities to better understand event relationships.

Independence

• Definition: Events A and B are independent if P(A|B) = P(A).

4.5 Bayes’ Theorem

• Importance: Provides a method to update initial or prior probabilities based on new evidence.

• Formula: P(A|B) = [P(A)P(B|A)] / P(B)

• Practical Example: Using Bayes’ Theorem to assess infection risk in livestock based on test results.

• General Application: Can accommodate multiple events, allowing a comprehensive analysis of scenarios where conditional probabilities need adjustment based on new information.

Application Example (L.S. Clothiers)

• Analyzed impacts of planning board recommendations on probabilities of town council decisions using Bayes’ theorem.