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 occurring in s to be |s|⁻¹ * |s ∩ t|. With this definition, we recover 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

• ProbabilityTheory.uniformOn: given a set s, uniformOn s is the counting measure conditioned on s. This is a probability measure when s is finite and nonempty.

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 uniformOn s P. We should avoid this however as none of the lemmas are written for predicates.

def ProbabilityTheory.uniformOn {Ω : Type u_1} [MeasurableSpace Ω] (s : Set Ω) : MeasureTheory.Measure Ω

Given a set s, uniformOn s is the uniform measure on s, defined as the counting measure conditioned by s. One should think of uniformOn s t as the proportion of s that is contained in t.

This is a probability measure when s is finite and nonempty and is given by ProbabilityTheory.uniformOn_isProbabilityMeasure.

Equations

• ProbabilityTheory.uniformOn s = MeasureTheory.Measure.count[|s]

instance ProbabilityTheory.instIsZeroOrProbabilityMeasureUniformOn {Ω : Type u_1} [MeasurableSpace Ω] {s : Set Ω} : MeasureTheory.IsZeroOrProbabilityMeasure (uniformOn s)

@[simp]

theorem ProbabilityTheory.uniformOn_empty_meas {Ω : Type u_1} [MeasurableSpace Ω] : uniformOn ∅ = 0

theorem ProbabilityTheory.uniformOn_empty {Ω : Type u_1} [MeasurableSpace Ω] {s : Set Ω} : (uniformOn s) ∅ = 0

@[simp]

theorem ProbabilityTheory.uniformOn_eq_zero' {Ω : Type u_1} [MeasurableSpace Ω] {s : Set Ω} (hs : MeasurableSet s) : uniformOn s = 0 ↔ s.Infinite ∨ s = ∅

See uniformOn_eq_zero for a version assuming MeasurableSingletonClass Ω instead of MeasurableSet s.

@[simp]

theorem ProbabilityTheory.uniformOn_eq_zero {Ω : Type u_1} [MeasurableSpace Ω] {s : Set Ω} [MeasurableSingletonClass Ω] : uniformOn s = 0 ↔ s.Infinite ∨ s = ∅

See uniformOn_eq_zero' for a version assuming MeasurableSet s instead of MeasurableSingletonClass Ω.

theorem ProbabilityTheory.finite_of_uniformOn_ne_zero {Ω : Type u_1} [MeasurableSpace Ω] {s t : Set Ω} (h : (uniformOn s) t ≠ 0) : s.Finite

theorem ProbabilityTheory.uniformOn_univ {Ω : Type u_1} [MeasurableSpace Ω] [Fintype Ω] {s : Set Ω} : (uniformOn Set.univ) s = MeasureTheory.Measure.count s / ↑(Fintype.card Ω)

theorem ProbabilityTheory.uniformOn_isProbabilityMeasure {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (hs : s.Finite) (hs' : s.Nonempty) : MeasureTheory.IsProbabilityMeasure (uniformOn s)

theorem ProbabilityTheory.uniformOn_singleton {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] (ω : Ω) (t : Set Ω) [Decidable (ω ∈ t)] : (uniformOn {ω}) t = if ω ∈ t then 1 else 0

theorem ProbabilityTheory.uniformOn_inter_self {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t : Set Ω} (hs : s.Finite) : (uniformOn s) (s ∩ t) = (uniformOn s) t

theorem ProbabilityTheory.uniformOn_self {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (hs : s.Finite) (hs' : s.Nonempty) : (uniformOn s) s = 1

theorem ProbabilityTheory.uniformOn_eq_one_of {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t : Set Ω} (hs : s.Finite) (hs' : s.Nonempty) (ht : s ⊆ t) : (uniformOn s) t = 1

theorem ProbabilityTheory.pred_true_of_uniformOn_eq_one {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t : Set Ω} (h : (uniformOn s) t = 1) : s ⊆ t

theorem ProbabilityTheory.uniformOn_eq_zero_iff {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t : Set Ω} (hs : s.Finite) : (uniformOn s) t = 0 ↔ s ∩ t = ∅

theorem ProbabilityTheory.uniformOn_of_univ {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (hs : s.Finite) (hs' : s.Nonempty) : (uniformOn s) Set.univ = 1

theorem ProbabilityTheory.uniformOn_inter {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t u : Set Ω} (hs : s.Finite) : (uniformOn s) (t ∩ u) = (uniformOn (s ∩ t)) u * (uniformOn s) t

theorem ProbabilityTheory.uniformOn_inter' {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t u : Set Ω} (hs : s.Finite) : (uniformOn s) (t ∩ u) = (uniformOn (s ∩ u)) t * (uniformOn s) u

theorem ProbabilityTheory.uniformOn_union {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t u : Set Ω} (hs : s.Finite) (htu : Disjoint t u) : (uniformOn s) (t ∪ u) = (uniformOn s) t + (uniformOn s) u

theorem ProbabilityTheory.uniformOn_compl {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (t : Set Ω) (hs : s.Finite) (hs' : s.Nonempty) : (uniformOn s) t + (uniformOn s) tᶜ = 1

theorem ProbabilityTheory.uniformOn_disjoint_union {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s t u : Set Ω} (hs : s.Finite) (ht : t.Finite) (hst : Disjoint s t) : (uniformOn s) u * (uniformOn (s ∪ t)) s + (uniformOn t) u * (uniformOn (s ∪ t)) t = (uniformOn (s ∪ t)) u

theorem ProbabilityTheory.uniformOn_add_compl_eq {Ω : Type u_1} [MeasurableSpace Ω] [MeasurableSingletonClass Ω] {s : Set Ω} (u t : Set Ω) (hs : s.Finite) : (uniformOn (s ∩ u)) t * (uniformOn s) u + (uniformOn (s ∩ uᶜ)) t * (uniformOn s) uᶜ = (uniformOn s) t

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