Axioms & Complements
Probability axioms.
Nonnegative, total=1, additive for disjoint events.
P(A∩B)=0
Mutually exclusive; handle independently.
Essence of conditional probabilities
Focus analysis to relevant context; refine judgment.
Exhaustive events.
Union covers full sample space.
Reliability as P(A)=0.9
High; sustain process, monitor drift.
Represent as 0≤P(E)≤1
Represents proportion; ensures logical consistency.
P(A∩B)>0
Overlap exists; coordinate responses.
P(A∩B)=P(A)×P(B)=0.1, causes this decision.
No conditional adjustment is required in prediction or risk assessment.
The need for exhaustiveness
Guarantees no missing outcomes; total probability=1.
P(A)=0.1
Rare but possible; include contingency.
Total probabilities > 1
Overlapping or incorrect event definitions.
P(A∪B)=P(A)+P(B) only possible
Only if disjoint.
P(A|B)>P(A)
B increases chance of A; positive
Event categories overlap
Double-count; inflated probability sum.
Decision framing for P(A)=0.5
Uncertain; collect data or defer choice.
Adding probabilities for disjoint events A and B
No overlap term.
High P(A∪B)
Wide exposure; strengthen joint mitigation.
P(B|A)=0 decision
A rules out B; mutually exclusive cases
Example: mutually exclusive but not exhaustive.
Pass/Fail ignoring pending results.
Subjective probability.
Belief-based estimate absent full data.
Sum of all event probabilities
1; total certainty across sample space.
P(A∩B)=P(A)P(B)
Independence; no effect between A,B.
Conditional probability key in quality testing answer in IJD - Interpretation, Judgement, Decision
I - links the observed evidence (like a sensor reading, defect detection, or pass/fail outcome) to the likelihood of the underlying cause.
J - Quantifies diagnostic reliability — how much confidence to place in what was observed.
D - P(Defect | Test Fail)), quality teams can decide which units to rework, when to recalibrate sensors, or how to tighten process control limits — ensuring that decisions are based on evidence, not assumptions.
Partition events into exhaustive sets
Supports total probability & Bayesian updating.
Communicate uncertainty
Improve trust, prevent false certainty.