Discrete Probability Ontology

FLOW ALGEBRA
TITLE: Discrete Probability Ontology
DOMAIN: Discrete Mathematics / Discrete Probability
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/discrete-probability/discrete-probability-ontology.html
VERSION: 1.0
DATE: 2026-10-11
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org
STATUS: Mathematical reference model; source page not independently verified.

1. ONTOLOGY PURPOSE
Model finite or countably infinite probability spaces, events, discrete random variables, distributions, conditional probabilities and derived statistical quantities with explicit mathematical constraints.

2. TYPED ENTITIES
entity SampleSpace Ω : nonempty finite_or_countable_set<Outcome>
entity Outcome ω : member(Ω)
entity Event A : subset(Ω)
entity ProbabilityMass p : function(Ω -> Real[0,1])
entity ProbabilityMeasure P : function(PowerSet(Ω) -> Real[0,1])
entity RandomVariable X : function(Ω -> CountableValueSet)
entity ProbabilityMassFunction p_X : function(ValueSet -> Real[0,1])
entity Distribution D_X : mapping(ValueSet, ProbabilityMass)
entity JointDistribution D_XY : mapping(ValuePair, ProbabilityMass)
entity ConditionalDistribution D_X_given_Y : mapping(Condition, Distribution)
entity Expectation E_X : extended_real_or_finite_real
entity Variance Var_X : nonnegative_extended_real
entity Independence : predicate(Event, Event)
entity Evidence : observation_or_proof
entity Claim : mathematical_statement

3. CORE SEMANTICS
P(A) := sum_{ω in A} p(ω)
forall ω in Ω: 0 <= p(ω) <= 1
sum_{ω in Ω} p(ω) = 1
P(Ω) = 1; P(empty_set) = 0
For pairwise disjoint events A_i: P(union_i A_i) = sum_i P(A_i)
p_X(x) := sum_{ω : X(ω)=x} p(ω)
sum_x p_X(x) = 1
P(X in B) := sum_{x in B} p_X(x)

4. FLOW OPERATIONS
flow EVENT_PROBABILITY(A:Event, p:ProbabilityMass) -> Real[0,1]:
  require A subset Ω and p is normalised
  return sum_{ω in A} p(ω)

flow CONDITIONAL(A:Event, B:Event) -> Real[0,1]:
  require P(B) > 0
  return P(A intersection B) / P(B)

flow TOTAL_PROBABILITY(A:Event, partition:{B_i}) -> Real[0,1]:
  require B_i are pairwise disjoint, union_i B_i = Ω
  return sum_i P(A|B_i)*P(B_i) for B_i with P(B_i)>0

flow BAYES(A:Event, B:Event) -> Real[0,1]:
  require P(B)>0 and P(A)>0
  return P(B|A)*P(A)/P(B)

flow EXPECTATION(X:RandomVariable) -> Real:
  require sum_x abs(x)*p_X(x) < infinity
  return sum_x x*p_X(x)

flow VARIANCE(X:RandomVariable) -> Real[0,infinity):
  require E[X^2] < infinity
  return E[X^2] - (E[X])^2

5. ALGEBRAIC LAWS
P(A complement) = 1 - P(A)
P(A union B) = P(A) + P(B) - P(A intersection B)
A subset B => P(A) <= P(B)
P(A intersection B) = P(A|B)*P(B) when P(B)>0
A independent B iff P(A intersection B) = P(A)*P(B)
E[aX+b] = aE[X]+b, when E[X] exists
Var(aX+b) = a^2 Var(X), when Var(X) exists
If X,Y have finite second moments: Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y)
If X,Y independent and integrable: E[XY]=E[X]E[Y]

6. STANDARD DISCRETE DISTRIBUTIONS
Bernoulli(q): support {0,1}; P(X=1)=q; E[X]=q; Var(X)=q(1-q)
Binomial(n,q): P(X=k)=C(n,k)q^k(1-q)^(n-k); 0<=k<=n
Geometric(q): support {1,2,...}; P(X=k)=(1-q)^(k-1)q; 0<q<=1
Poisson(λ): P(X=k)=exp(-λ)λ^k/k!; λ>0; k>=0
UniformFinite(S): P(X=x)=1/|S| for x in nonempty finite S

7. RELATIONSHIPS
DiscreteProbability IS_A ProbabilityTheory
ProbabilityMass DEFINES ProbabilityMeasure
Event IS_SUBSET_OF SampleSpace
RandomVariable MAPS Outcome TO Value
RandomVariable INDUCES ProbabilityMassFunction
JointDistribution DETERMINES MarginalDistribution
ConditionalProbability REQUIRES PositiveConditionProbability
Independence IS_DEFINED_BY ProductFactorisation
Expectation SUMMARISES Distribution
Variance QUANTIFIES Dispersion

8. INFERENCE RULES
rule R1: A subset B => P(A)<=P(B)
rule R2: disjoint(A,B) => P(A union B)=P(A)+P(B)
rule R3: independent(A,B) and P(B)>0 => P(A|B)=P(A)
rule R4: valid_partition(B_i) => P(A)=sum_i P(A|B_i)P(B_i) for positive-mass B_i
rule R5: P(A)>0 and P(B)>0 => P(A|B)=P(B|A)P(A)/P(B)
rule R6: distribution_normalised(p_X) => sum_x p_X(x)=1

9. VALIDATION
validate nonempty(Ω)
validate every event subset Ω
validate every p(ω) finite and in [0,1]
validate abs(sum_ω p(ω)-1) <= numerical_tolerance
validate conditional denominators strictly positive
validate countable sums converge to required values
validate finite expectation and variance before using finite formulas
validate support and parameter restrictions for named distributions
validate mathematical proof claims separately from empirical probability estimates

10. SEMANTIC_EQUIVALENCE
'probability mass function' <=> 'PMF'
'discrete random variable distribution' <=> 'discrete probability distribution' (when referring to its induced law)
'expected value' <=> 'expectation'
'events are independent' <=> 'joint probability equals product of marginals'
Note: 'mutually exclusive' is NOT equivalent to 'independent' except in degenerate zero-probability cases.

11. CONFIDENCE
formal_definition: 1.00 when axioms and assumptions satisfied
proved_identity: 1.00 when derivation verified
computed_probability: dependent_on_input_validity_and_numerical_error
empirical_estimate: requires sample_size, method, uncertainty and provenance
unsupported_claim: unverified

12. FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8; LF line endings; stable ordering of typed entities and rules
scope: canonical semantic payload excluding computed FLOW_HASH value
value: COMPUTE_AT_INGESTION (not precomputed)

13. UPDATE_MODEL
version_policy: semantic_versioning
on_change: revalidate axioms, formulas, examples, and dependent ontology links
provenance: retain source URL, revision timestamp and change summary
breaking_change: modified entity types, probability conventions or rule semantics

14. AI_CONSUMPTION
input: structured mathematical question or finite/countable probability model
pipeline: parse -> type_check -> validate_assumptions -> select_rule -> derive -> verify -> explain
output: result, assumptions, derivation, confidence, provenance, unresolved_constraints
safety: never infer independence without evidence; reject division by zero; distinguish model probabilities from observed frequencies

15. WORKED EXAMPLE
Ω = {1,2,3,4,5,6}; p(ω)=1/6
A = {2,4,6}; B = {4,5,6}
P(A)=3/6=1/2
P(B)=3/6=1/2
P(A intersection B)=2/6=1/3
P(A|B)=(1/3)/(1/2)=2/3
A and B are not independent because 1/3 != (1/2)*(1/2).

END FLOW ALGEBRA