Joint & Marginal Probabilities
Joint probability.
Chance that A and B occur together (A∩B).
Law of Total Probability
Use it when estimating overall risk, defect rates, or success probabilities across multiple pathways or subgroups.
Bayes’ theorem in words.
Updates prior belief after observing new evidence.
Use probability models in management
Quantify uncertainty for rational policies.
Measurement approaches Observation
Repeated observation approximates true value within noise
Marginal probability.
Single-event probability, summing across others.
Partition the sample space before applying the Law of Total Probability?
Guarantees a complete and accurate calculation of overall probability or risk by combining all valid subcases.
Bayesian reasoning purpose
Adjust expectations rationally with data.
The risk in modeling or policy decisions if the assumption of independence between events does not hold (IJD)
Interpretation:
When events are not truly independent, their combined probability cannot be obtained by simple multiplication — real outcomes may co-occur more often than the model predicts.
Judgment:
Ignoring this dependence leads to underestimating joint risks or failure rates.
Decision:
Re-evaluate models and policies to include correlation or interaction effects; otherwise, plans may be overly optimistic and safeguards inadequate.
Variability in relation to measurement count
Variability decreases as number of measurements increases
P(A∩B)=0.15,P(A)=0.3,P(B|A) = 0.5
B occurs half of times with A.
P(A₁)=0.6, P(A₂)=0.4, P(B|A₁)=0.2, and P(B|A₂)=0.5, The combined probability P(B)=0.32 tells us.
Use this total probability to plan overall system performance, resource allocation, or defect prediction — not just within one subset, but across the entire process mix.
Example: 90% accurate test, 10% defect rate, P(defect|positive).
Posterior moderate; many positives false—verify before action.
Validate models
Ensure predictions match observed frequencies.
Weight estimation by variability
Lower-variance observations get higher confidence in inference.
P(A∩B)<<P(A)P(B)
Negative relation; co-occurrence rare.
Total probability of event B is 0.32 and quality assessment
Prioritize control, monitoring, or improvement efforts on that higher-risk condition (A₂) to reduce the overall probability of B.
Bayesian updating mislead may mislead
Prior or likelihood unreliable, inference biased.
The use of simulation in Probability models and policy.
Reveal range of likely outcomes under uncertainty.
Bayesian thinking conceptually.
Beliefs (priors) updated by evidence to form new (posterior) understanding.
The conditional probability P(A | B) equals the unconditional probability P(A)
Independent. Treat them separately in modeling or policy decisions; no adjustment or correction is needed when one occur
Total probability in forecasting
Aggregates conditional probabilities into total expectation.
Apply Bayesian updating in predictive maintenance.
Combines prior fault rates with sensor data to refine predictions.
This is an example of probability models and policy metric.
SLA = P(meeting demand)≥target.
Bayesian thinking and its importance in data science and AI.
Drives learning from data—models adjust confidence as evidence accumulates for better automated decisions.