Check point 1: Basic Set Theory and Probability Definition

Random Experiment

A phenomenon with an unknown outcome.

Element

A possible outcome of an experiment.

Sample Space (S)

The set of all possible outcomes of an experiment.

Event (E)

A set containing possible outcomes of an experiment, denoted E\subseteq S.

E Complement

The complement of an event E, denoted as E^c, which consists of all outcomes in sample space S that are not in E.

Union (AND)

A set consisting of all outcomes found in any of the events.

Intersection (OR)

A set consisting of only the common outcomes of any of the events.

Mutually Exclusive (Disjoint)

Events that have no common outcomes, denoted E\cap F = 0.

Bayes’ Formula

P(B|A) = \frac{P(A|B)P(B)}{P(A)}.

Independence Check

If P(A|B) = P(A), then events A and B are independent.

Complement Independence

If P(A) and P(B) are independent, then P(A) and P(B^c) are also independent.

Conditional Probability

P(A | B) = \frac{A\cap B}{B}.

Law of Total Probability

Probability of an event can be calculated by considering all the different ways it could happen.

Counting

How many ways something can happen, calculated by multiplying the number of stages by the possibilities of each stage.

Permutation

An ordered arrangement of items.

Permutation Formula

n! = n(n-1)(n-2)…3×2×1.