mathlib documentation

probability.cond_count

Classical probability #

The classical formulation of probability states that the probability of an event occurring in a finite probability space is the ratio of that event to all possible events. This notion can be expressed with measure theory using the counting measure. In particular, given the sets s and t, we define the probability of t occuring in s to be |s|⁻¹ * |s ∩ t|. With this definition, we recover the the probability over the entire sample space when s = set.univ.

Classical probability is often used in combinatorics and we prove some useful lemmas in this file for that purpose.

Main definition #

Notes #

The original aim of this file is to provide a measure theoretic method of describing the probability an element of a set s satisfies some predicate P. Our current formulation still allow us to describe this by abusing the definitional equality of sets and predicates by simply writing cond_count s P. We should avoid this however as none of the lemmas are written for predicates.

noncomputable def probability_theory.cond_count {Ω : Type u_1} [measurable_space Ω] (s : set Ω) :

Given a set s, cond_count s is the counting measure conditioned on s. In particular, cond_count s t is the proportion of s that is contained in t.

This is a probability measure when s is finite and nonempty and is given by probability_theory.cond_count_is_probability_measure.

Equations
theorem probability_theory.cond_count_singleton {Ω : Type u_1} [measurable_space Ω] [measurable_singleton_class Ω] (ω : Ω) (t : set Ω) [decidable t)] :
theorem probability_theory.cond_count_eq_one_of {Ω : Type u_1} [measurable_space Ω] [measurable_singleton_class Ω] {s t : set Ω} (hs : s.finite) (hs' : s.nonempty) (ht : s t) :

A version of the law of total probability for counting probabilites.