Probability Fundamentals & Bayes

Probability quantifies uncertainty. Learn experiments, sample spaces and events, the axioms, complements, unions and intersections, conditional probability, independence, the multiplication rule, total probability, Bayes' theorem, and reliability of series and parallel systems.

Component failures, defect rates, communication errors, sensor false alarms and load uncertainties are all probabilistic. Engineers use probability to design reliable systems and interpret test results correctly.

An analogy

Probability is like the long-run fraction of times something happens if you could repeat the experiment endlessly — a coin lands heads about half the time.

Events and counting

For equally likely outcomes P(A) = favourable/total. Complement rule: P(at least one) = 1 − P(none) — often the easiest route.

Conditional probability and independence

P(A|B) updates probability given information. Independence means knowing B tells nothing about A. Components sharing a power supply or environment are often NOT independent (common-cause failures).

Bayes' theorem

Reverses conditioning: from P(test positive | defect) to P(defect | test positive). With rare defects, even accurate tests give many false positives (base-rate fallacy).

System reliability

Series system works only if all work: R = ΠRᵢ. Parallel (redundant) fails only if all fail: R = 1 − Π(1 − Rᵢ). Complex networks use decomposition, fault trees and Monte Carlo.

Formulas

• Axioms: 0 ≤ P(A) ≤ 1, P(S) = 1, P(A ∪ B) = P(A) + P(B) for disjoint A, B
• Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• Conditional: P(A|B) = P(A ∩ B)/P(B)
• Independence: P(A ∩ B) = P(A)P(B)
• Total probability: P(B) = Σ P(B|Aᵢ)P(Aᵢ)
• Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B)

Worked example: Bayes: inspection test

Given: Defect rate 2 %, test sensitivity 95 %, false-positive rate 5 %

Find: P(defective | positive).

• P(+) = 0.02 × 0.95 + 0.98 × 0.05 = 0.068
• P(D|+) = 0.019/0.068 = 0.279

Answer: ≈ 28 % — most alarms are false

Worked example: Reliability

Given: Three components R = 0.95 in series; two R = 0.9 in parallel

Find: System reliabilities.

• Series: 0.95³ = 0.857
• Parallel: 1 − 0.1² = 0.99

Answer: 0.857 and 0.99

Common mistakes

• Wrong: Confusing P(A|B) with P(B|A). — Right: Use Bayes to reverse conditioning.
• Wrong: Assuming independence of common-cause failures. — Right: Shared environments correlate failures.

Applications

• Quality inspection: Interpreting test alarms with Bayes.
• Reliability engineering: Redundancy design.
• Risk assessment: Fault and event trees.

Key takeaways

• Axioms, complements, addition rule.
• Conditional probability and independence.
• Bayes reverses conditioning — beware base rates.
• Series R = ΠRᵢ; parallel 1 − Π(1 − Rᵢ).

Going further

• Fault tree analysis: AND/OR gates for top events.
• Bayesian updating: Posterior distributions of parameters.
• Markov models: Repairable system availability.