Probability quantifies uncertainty and underpins econometrics.
It serves as a foundational topic for the course, indicating its importance for succeeding content throughout the semester.
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}
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.
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.
For a 6-sided die: Each outcome has a probability of 1/6.
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.
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.
Calculate probabilities in scenarios like getting a flat tire and overheating the car.
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.
Concept: Displays joint and marginal probabilities to better understand event relationships.
Definition: Events A and B are independent if P(A|B) = P(A).
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.
Analyzed impacts of planning board recommendations on probabilities of town council decisions using Bayes’ theorem.