Learning Objectives
- Understand sample space and events
- Learn axiomatic approach to probability
- Study types of events and their probabilities
- Apply addition theorem and conditional concepts
Key Concepts
Random Experiment and Sample Space
A random experiment is one whose outcome cannot be predicted with certainty. Sample space (S): Set of all possible outcomes. Example: Tossing a coin: S = {H, T}. Rolling a die: S = {1, 2, 3, 4, 5, 6}. Two coins: S = {HH, HT, TH, TT}.
Events
An event is a subset of sample space. Simple event: Single outcome. Compound event: More than one outcome. Sure/Certain event: S itself (P(S) = 1). Impossible event: φ (P(φ) = 0). Complementary event: A' = S - A; P(A') = 1 - P(A).
Mutually exclusive events: A ∩ B = φ (cannot occur simultaneously). Example: Getting head and tail in a single coin toss. Exhaustive events: Their union is S. Independent events: Occurrence of one doesn't affect the other.
Axiomatic Approach
Kolmogorov's axioms: (1) P(E) ≥ 0 for any event E. (2) P(S) = 1. (3) For mutually exclusive events E₁, E₂, ...: P(E₁ ∪ E₂ ∪ ...) = P(E₁) + P(E₂) + ... . For equally likely outcomes: P(A) = n(A)/n(S) = favorable outcomes / total outcomes.
Addition Theorem
P(A ∪ B) = P(A) + P(B) - P(A ∩ B). If mutually exclusive: P(A ∪ B) = P(A) + P(B). For three events: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) - P(A∩C) + P(A∩B∩C).
P(A but not B): P(A) - P(A∩B) = P(A ∩ B'). P(neither A nor B): P(A' ∩ B') = 1 - P(A ∪ B) = P((A ∪ B)').
Important Results
0 ≤ P(A) ≤ 1. P(A') = 1 - P(A). P(A ∪ B) ≤ P(A) + P(B). If A ⊆ B, then P(A) ≤ P(B). P(A ∩ B') + P(A' ∩ B) = P(A) + P(B) - 2P(A ∩ B).
Summary
Probability quantifies the likelihood of events. The sample space contains all possible outcomes. Events are subsets of the sample space. The axiomatic approach defines probability satisfying non-negativity, normalization, and additivity. The addition theorem handles union of events.
Important Terms
- Sample space: Set of all possible outcomes of an experiment
- Event: Subset of sample space
- Mutually exclusive: Events that cannot occur together (A ∩ B = φ)
- Complementary event: A' = S - A; P(A') = 1 - P(A)
- Addition theorem: P(A∪B) = P(A) + P(B) - P(A∩B)
- Equally likely: All outcomes have equal probability
Quick Revision
- P(A) = favorable outcomes / total outcomes (equally likely)
- 0 ≤ P(A) ≤ 1; P(S) = 1; P(φ) = 0
- P(A') = 1 - P(A)
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
- Mutually exclusive: P(A ∪ B) = P(A) + P(B)
- Two dice: S has 36 outcomes; Two coins: 4 outcomes
- P(at least one) = 1 - P(none)
- P(exactly one of A,B) = P(A) + P(B) - 2P(A∩B)