discrete probability introduction

discrete probability

probability where the outcome is a member of a defined set of values e.x. rolling a die {1, 2, 3, 4, 5, 6} or flipping a coin {H, T}

continuous probability

probability where the outcome can have a continuous set of values e.x. height

sample space

notated Ω the set of all possible outcomes of an experiment

outcome notation

ω outcomes are also known as elementary events

event

a set of outcomes that is subset of the sample space

probability of an event

the probability of event A is the probability of outcome omega (Ω), such that omega is a member of the set of outcomes A, where set A is a subset of the sample space can be notated more simply as P(A)

numeric representation of probability

probability is measured as a number between 0 and 1 (inclusive) P(∅) = 0 (∅ is no outcome) P(Ω) = 1

finite discrete probability space

notated (Ω, P) the sample space tohether with a probability distribution (or probability mass function) function P : Ω -> R, that maps the set of outcomes to a real number

combination notation

for events A and B and: (EA ∧ EB) = P(A ∩ B) or: P(EA ∨ EB ) = P(A ∪ B) not A: P(¬EA) = P(Ā)

probability of AND

P(A ∩ B) = P(A)P(B) only works on independent events

probability of OR

P(A ∪ B) = P(A) + P(B) only works on disjoint events (events that cannot happen at the same time) for non-disjoint events P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

conditional probability

when the probability of an event is affected by the outcome of a previous event probability of B given A is notated P(B|A)

conditional probability calculations

P(B ∩ A) = P(B|A)P(A) P(B|A) = P(B ∩ A)/P(A) P(A) must not be zero

bayes' rule

P(B|A) = P(A|B)P(B)/P(A) P(A) must not be zero

bayes' rule terminology

P(B) refers to the prior (overall, independent) probability of B P(A|B) is the likelihood function given the probability that A has already happened given B P(B|A) is the posterior probability of B given A