Stats U5

Gambler’s Fallacy

The belief that past events can influence the probability of future events in random processes, leading one to erroneously expect a certain outcome based on previous occurrences.

Axiom 1 of probability

If A is an event then its probability P(A) is between 0 and 1 —> 0 ≤ P(A) ≤ 1

Axiom 2 of Probability

If s is the sample space, then P(s) = 1

Axiom 3 of Probability

if 2 events A & B are disjoint (mutually exclusive) [A∩B = 0] then P(A or B) = P(A) + P(B)

Complement

The complement of A, denoted as A^C is the event that A does not occur

Intersection

The intersection of events A and B is the event that both occur, denoted as ( A ∩ B )

Union

Event that A or B occurs (inclusive), or both occur denoted as (A ∪ B)

DeMorgan Law

A set of rules that describes how the complement of the union and intersection of sets relates to each other. Specifically, it states that the complement of the union of two sets is equal to the intersection of their complements, and vice versa.

dependent events

if 2 events A and B are mutually exclusive (disjoint) or if P(A | B) ≠ P(A)

Law of Total Probability

P(B) = P(B | A)P(A) + P(B | A^C)P(A^C) states that the probability of an event B can be found by considering all the possible ways that B can occur in conjunction with another event A and its complement.

Bayes rule

A fundamental theorem in probability that describes how to update the probability of a hypothesis based on new evidence. It is expressed as $P(A | B) = \frac{P(B | A)P(A)}{P(B)}$, where events A and B are relevant to each other.