Probability and Stats

Set

A ⊆ B : A is a subset of B if all elements of A are also members of B

Cardinality

The measure of the number of elements of a set. A set with a single element is a singleton

Important sets

∅ is a subset of A for all A ≠ ∅

Set operations

A∪B : the union of two sets contains all elements of only A, only B and the elements shared by both

A∩B : the intersection of two sets contains elements found in both A and B

Properties of the union

• If a collection of sets are disjoint but the union of the sets is the universal set, then the sets form a partition of the universal set

• A∪A’ is the universal set

Properties of the intersection

A∩A’ is the empty set

Distributive law

De Morgan’s laws

Sample spaces

• the outcome of a random experiment is unknown but all possible outcomes can be described by the sample space

• Events are subsets of the sample space

• The empty set is the null event - it never occurs

• The singleton subset of S is called an elementary event of S

• The sample space S consists of the union of all elementary events called the universal event

Probability axioms

Boole’s Relationship

Conditional Probability

• The probability of an event occurring given that another event has already occurred

• If A is a subset of B, P(A | B) = 1

• If A and B don’t intersect, P(A | B) = 0

• Events A and B are said to be independent if P(A|B) = P(A)

• P(A’ | B) = 1 - P(A|B)

Multiplication Law of Probability

• For independent events : P(A∩B) = P(A|B)P(B) = P(A)P(B)

Probability Table

Total Probability

• you can find the probability of an event B by splitting it up according to which of several mutually exclusive scenarios A1…Ak happened, then adding up their contributions

• Note: P(B|Ai)P(Ai) = P(B∩Ai)

Bayes Theorem