Basics of Probability Theory

Practice flashcards covering the fundamental concepts, definitions, and axioms of probability theory as presented by Ing. Tomáš Urbánek, Ph.D.

Random Experiment

An experiment or situation that can have different outcomes and should ideally be repeatable any number of times.

Random Event (A)

Every possible result of a random experiment; a specific situation or outcome that can occur.

Sample Space ($\Omega$)

The set or collection of all possible outcomes of a random experiment.

Impossible Event

An event with a probability of $0$.

Certain Event

An event with a probability of $1$.

Probability ($P(A)$)

A measure between $0$ and $1$ expressing how likely an event is, calculated as the ratio of favorable cases to total possible cases.

Classical Probability

A type of probability used when all possible outcomes of a random experiment are equally likely, defined by the formula $P(A) = \frac{m}{n}$.

Statistical Probability

Probability derived from repeated experiments, empirical data, or observations, defined as the limit of frequency: $P(A) = \lim_{n \to \infty} \frac{m}{n}$.

Andrej Nikolajevič Kolmogorov

Soviet mathematician and founder of modern probability theory who established the axiomatic definition of probability.

Kolmogorov's First Axiom

An axiom stating that the probability of an event is a real number greater than or equal to zero: $P(A) \in \mathbb{R}$, $P(A) \geq 0$.

Kolmogorov's Second Axiom

An axiom stating that the probability of the entire sample space is equal to one: $P(\Omega) = 1$.

Kolmogorov's Third Axiom

An axiom stating that for a sequence of mutually disjoint events $A_1, A_2, \dots$, the probability of their union is the sum of their individual probabilities: $P(\bigcup_{i=1}^{\infty} A_i) = \sum_{i=1}^{\infty} P(A_i)$.

Complementary Event ($\bar{A}$)

The set of all outcomes in the sample space that are not in event $A$, such that $P(A) + P(\bar{A}) = 1$.

Probability of Union of Non-Disjoint Events

The rule used when events $A$ and $B$ have an intersection ($P(A \cap B) \neq \emptyset$), defined as $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.