Unit 2: Elementary Probability Theory
Course: Statistics for Economics (ECODSC 253)
Probability theory provides the mathematical framework for quantifying uncertainty, which is essential for economic forecasting and decision-making under risk.
Table of Contents
1. Random Experiment, Sample Spaces, and Events
A Random Experiment is a process where the exact outcome cannot be predicted with certainty, even if the experiment is repeated under the same conditions.
Sample Space (S)
The set of all possible outcomes of a random experiment. For example, in a coin toss, S = {Heads, Tails}.
Events
An Event is a subset of the sample space.
- Simple Event: An event with only one outcome.
- Compound Event: An event with more than one outcome.
- Mutually Exclusive Events: Events that cannot happen at the same time.
2. Probability Axioms and Properties
Modern probability is built on three fundamental axioms:
- Non-negativity: The probability of any event A is always greater than or equal to zero (P(A) ≥ 0).
- Certainty: The probability of the entire sample space S is 1 (P(S) = 1).
- Additivity: For mutually exclusive events, the probability of their union is the sum of their individual probabilities.
Property: The probability of an event always lies between 0 and 1 (0 ≤ P(A) ≤ 1).
3. Addition and Multiplication Theorems
Addition Theorem
Used to find the probability of at least one of two events occurring (A or B).
If A and B are mutually exclusive, P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
Multiplication Theorem
Used to find the probability of two events occurring together (A and B).
4. Conditional Probability and Independence
Conditional Probability is the probability of event A occurring given that event B has already occurred.
Independence of Events
Two events A and B are Independent if the occurrence of one does not affect the probability of the other.
Condition for Independence: P(A ∩ B) = P(A) * P(B)
5. Bayes Theorem
Bayes Theorem describes the probability of an event based on prior knowledge of conditions that might be related to the event. It is a way to "reverse" conditional probabilities.
In economics, this is often used to update forecasts as new data (B) becomes available for various economic scenarios (Ai).
Exam Corner: Common Pitfalls
- Mutually Exclusive vs Independent: These are NOT the same. If events are mutually exclusive, they cannot be independent (if one happens, the other definitely won't).
- Total Probability: Always check if the denominator in Bayes theorem (the total probability) sums up all possible mutually exclusive paths.
- Formula Tip: P(A) + P(Not A) = 1. Using the complement is often easier than calculating complex events directly.