Probability Diagnostic Framework
Core Formulas
- P(A) = Favorable / Total outcomes
- P(A ∩ B) = P(A) × P(B|A) (joint)
- P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- P(A|B) = P(A ∩ B) / P(B) (conditional)
- Bayes: P(A|B) = P(B|A)·P(A)/P(B)
Event Types
- Independent: P(A ∩ B) = P(A)·P(B)
- Mutually exclusive: P(A ∩ B) = 0
- Complement: P(not A) = 1 − P(A)
Rules & Heuristics
- Sum of all probabilities = 1
- For “at least one”: 1 − P(none)
- For “exactly k”: Binomial(n, k, p)
- Replacement matters: with → independent, without → dependent
Common Pitfalls
- Don’t confuse independence with exclusivity
- Check denominator: never zero, always all possible outcomes
- Conditional ≠ reverse conditional (P(A|B) ≠ P(B|A))
FAQ
- Can probability be negative or over 1? No, always 0 ≤ P ≤ 1
- What if events overlap? Use inclusion-exclusion: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- How to check logic? All probabilities must sum to 1; check edge cases
Action Tip: Always define your sample space and event clearly before plugging into formulas.