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.

1. Random Experiment, Sample Spaces, and Events
2. Probability Axioms and Properties
3. Addition and Multiplication Theorems
4. Conditional Probability and Independence
5. Bayes Theorem

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).

P(A ∪ B) = P(A) + P(B) - P(A ∩ 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).

P(A ∩ B) = P(A) * P(B|A)

4. Conditional Probability and Independence

Conditional Probability is the probability of event A occurring given that event B has already occurred.

P(A|B) = P(A ∩ B) / P(B), where P(B) > 0

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.

P(Ai|B) = [P(Ai) * P(B|Ai)] / [Σ P(Ak) * P(B|Ak)]

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.