Probability for Discrete Variables

Vocabulary flashcards covering core definitions, probability rules, sampling types, and formulas for discrete random variables.

Pierre-Simon Laplace

The scholar who remarked that a science which began with the consideration of games of chance has become the most important object of human knowledge.

Experiment (Probability)

An action that gives rise to an event or set of events.

Sample Space ($S$)

The set that contains all possible events or outcomes of an experiment, satisfying $P(S) = 1$.

Event / Outcome ($A$)

A specific result or collection of results occurring within a sample space, bounded by the rule $0 \bowtie P(A) \bowtie 1$.

Marginal (Unconditional) Probability

The probability of a single discrete event occurring, denoted $P(A)$. For $N$ equally probable outcomes, it equals $P(A_i) = \frac{N_{A_i}}{N}$.

Complement of an Event

The probability of an event not occurring, defined mathematically as $P(A_i') = 1 - P(A_i)$.

Joint Probability (Intersection, $A \cap B$)

The probability of two events $A$ and $B$ occurring together simultaneously.

Union ($A \cup B$)

The probability of the occurrence of either event $A$ or event $B$ (the occurrence of any event).

Conditional Probability ($A|B$)

The probability of event $A$ occurring conditional on the occurrence of event $B$.

Mutually Exclusive Events

Events that cannot happen at the same time, meaning their joint probability is zero ($P(A \cap B) = 0$) and $P(A|B) = 0$.

Independent Events

Events where the occurrence of event $A$ provides no information about the occurrence of event $B$ (and vice versa), satisfying $P(A|B) = P(A)$.

Dependent Events

Events where the occurrence of event $A$ tells us something about the occurrence of event $B$ by affecting its probability of occurring.

Sampling Without Replacement

A sampling strategy where selected items are not returned to the pool, thereby creating dependencies between trials when sampling from a finite sample space.

Additive Rule of Probability

The principle stating that if two discrete events, $A$ and $B$, are mutually exclusive, then $P(A \cup B) = P(A) + P(B)$.

Multiplicative Rule of Probability

The principle stating that if two discrete events, $A$ and $B$, are independent, then $P(A \cap B) = P(A) \times P(B)$.

Conditional Probability Formula for Dependent Events

The formula used to calculate conditional probability when events are dependent: $P(A|B) = \frac{P(A \cap B)}{P(B)}$.

Probability Mass Function (PMF)

A function that gives the probability of each distinct outcome for a discrete variable, written as $P(X = x_i)$ for $i = 1, 2, 3, \dots, N$.