Definition of Events
Event Concept
Event: subset of sample space representing one or more outcomes. Occurs if experiment outcome lies in this subset. Basis of probability measurement.
Relation to Experiment
Experiment: process with uncertain results. Event depends on experiment's outcome. Events characterize possible results.
Notation
Events denoted by capital letters: A, B, C, etc. Sample space: S or Ω. Event A ⊆ S.
Types of Events
Simple Events
Contain single outcome. Indivisible. Example: rolling a 3 on a die.
Compound Events
Contain multiple outcomes. Formed by union of simple events. Example: rolling an even number.
Sure Event
Event containing all outcomes. Denoted S. Probability = 1.
Impossible Event
Empty set with no outcomes. Probability = 0.
Sample Space and Outcomes
Sample Space Definition
Set of all possible outcomes of an experiment. Denoted S or Ω.
Outcomes
Individual possible results. Elements of S. Example: {1,2,3,4,5,6} for die.
Discrete and Continuous
Discrete: countable outcomes. Continuous: uncountable, intervals.
Simple Events
Definition
Events with exactly one outcome. Atomic events in probability space.
Examples
Flipping head on coin: {H}. Drawing ace of spades: single card event.
Probability
Sum of probabilities of all simple events = 1. Each simple event assigned probability.
Compound Events
Definition
Events formed by combination of simple events. Union, intersection, or complement.
Union of Events
A ∪ B: event that either A or B or both occur.
Intersection of Events
A ∩ B: event that both A and B occur simultaneously.
Complement of Event
A': event that A does not occur. A' = S \ A.
Mutually Exclusive Events
Definition
Events that cannot occur simultaneously. A ∩ B = ∅.
Examples
Rolling a 3 and rolling a 5 on one die roll.
Probability Implication
For mutually exclusive events A and B: P(A ∪ B) = P(A) + P(B).
Independent Events
Definition
Events where occurrence of one does not affect probability of other.
Mathematical Condition
P(A ∩ B) = P(A) × P(B).
Examples
Flipping two coins: outcome of one does not affect the other.
Operations on Events
Union (OR)
A ∪ B: event either A or B or both occur. Combines outcomes.
Intersection (AND)
A ∩ B: event both A and B occur together.
Complement (NOT)
A': event A does not occur; complement of A in S.
Difference
A \ B: outcomes in A but not in B.
Event Probability
Definition
Probability of event A: measure of likelihood A occurs. P(A) ∈ [0,1].
Calculation
Sum of probabilities of outcomes in A.
Properties
P(S) = 1, P(∅) = 0, 0 ≤ P(A) ≤ 1.
Probability Axioms
Non-negativity, normalization, countable additivity.
Conditional Probability
Definition
Probability of A given B occurred: P(A|B).
Formula
P(A|B) = P(A ∩ B) / P(B), provided P(B) > 0.
Interpretation
Updates probability of A based on occurrence of B.
Law of Addition
General Formula
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
For Mutually Exclusive Events
If A and B disjoint: P(A ∪ B) = P(A) + P(B).
Extension
Generalized to multiple events with inclusion-exclusion principle.
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)Law of Multiplication
General Formula
P(A ∩ B) = P(A) × P(B|A).
For Independent Events
P(A ∩ B) = P(A) × P(B).
Extension
Extended for multiple events: P(A ∩ B ∩ C) and so on.
P(A ∩ B) = P(A) × P(B|A)Examples and Applications
Example 1: Rolling a Die
Event A: rolling an even number = {2,4,6}. P(A) = 3/6 = 0.5.
Example 2: Coin Tosses
Events A: first coin heads, B: second coin heads. Independent. P(A ∩ B) = 0.5 × 0.5 = 0.25.
Example 3: Drawing Cards
Event A: drawing a heart; P(A) = 13/52 = 0.25. Event B: drawing an ace; P(B) = 4/52 = 0.077.
Event Table for Card Example
| Event | Description | Probability |
|---|---|---|
| A | Draw a heart | 13/52 = 0.25 |
| B | Draw an ace | 4/52 ≈ 0.077 |
| A ∩ B | Draw ace of hearts | 1/52 ≈ 0.019 |
Interpretation
Calculations demonstrate event composition, intersection probability, and event dependency.
References
- Feller, W. Introduction to Probability Theory and Its Applications, Wiley, Vol. 1, 1968, pp. 1-500.
- Ross, S. M. A First Course in Probability, Pearson, 10th Ed., 2018, pp. 1-720.
- Grinstead, C. M., Snell, J. L. Introduction to Probability, American Mathematical Society, 1997, pp. 1-300.
- Billingsley, P. Probability and Measure, Wiley, 3rd Ed., 1995, pp. 1-600.
- Durrett, R. Probability: Theory and Examples, Cambridge University Press, 5th Ed., 2019, pp. 1-500.