Probability Rules
Probability quantifies likelihood on a scale from 0 (impossible) to 1 (certain). The key rules — addition, multiplication, and conditional probability — handle any event combination.
Basic Rules
Complement: P(A') = 1 - P(A)
Addition: P(A∪B) = P(A) + P(B) - P(A∩B)
Mutually exclusive: P(A∪B) = P(A) + P(B)
Multiplication P(A∩B) = P(A) × P(B|A)
Independent: P(A∩B) = P(A) × P(B)Conditional Probability
P(B|A) = P(A∩B) / P(A)
"Probability of B given A has occurred"
Card example: P(Ace | Red card)
= P(Red Ace) / P(Red) = (2/52) / (26/52) = 2/26 = 1/13Bayes' Theorem
P(A|B) = P(B|A) × P(A) / P(B)
Medical test: disease prevalence 1%, test 99% accurate
P(disease|positive) = (0.99×0.01) / [(0.99×0.01)+(0.01×0.99)]
= 0.0099/0.0198 = 50%
(Even with 99% accuracy, low prevalence → many false positives)Calculate probabilities: Free Probability Calculator
Probability Rules Quick-Reference Table
| Rule | Formula | When to use |
|---|---|---|
| Addition (OR, mutually exclusive) | P(A∪B) = P(A) + P(B) | Events can't both occur |
| Addition (OR, general) | P(A∪B) = P(A) + P(B) − P(A∩B) | Events can both occur |
| Multiplication (AND, independent) | P(A∩B) = P(A) × P(B) | Independent events |
| Multiplication (AND, dependent) | P(A∩B) = P(A) × P(B|A) | Dependent events |
| Complement | P(A') = 1 − P(A) | "At least one" problems |
| Bayes' Theorem | P(A|B) = P(B|A)P(A) / P(B) | Updating beliefs with evidence |
How Probability Works
Probability measures the likelihood of an event on a scale from 0 (impossible) to 1 (certain). Classical probability: P(A) = (favourable outcomes) / (total equally likely outcomes). Frequentist interpretation: long-run relative frequency in repeated experiments. Bayesian interpretation: degree of belief, updated with evidence via Bayes' theorem.
Combinatorics underpins counting outcomes: permutations P(n,r) = n!/(n−r)! for ordered selections; combinations C(n,r) = n!/[r!(n−r)!] for unordered. The birthday problem is a famous counter-intuitive result: in a group of just 23 people, there is a greater than 50% chance that two share a birthday — because you are comparing 253 pairs, not 22 individual matches against one birthday.
Common Mistakes
- Assuming independence incorrectly: Drawing cards without replacement makes events dependent. Failure rates in systems with shared components are correlated, not independent. Always verify independence before multiplying probabilities.
- Confusing P(A|B) with P(B|A): The base rate fallacy: a test with 99% sensitivity (P(positive|disease)) does not mean a positive test means 99% chance of disease — the prior probability P(disease) matters enormously. This is Bayes' theorem in action.
- Neglecting the complement: "At least one" problems are much easier via complement: P(at least one success in n trials) = 1 − P(zero successes) = 1 − (1−p)ⁿ.
Frequently Asked Questions
You choose one of 3 doors (car behind one; goats behind two). The host opens a goat door. Should you switch? Yes — switching wins ²/₃ of the time. After your initial choice, there is a ²/₃ chance the car is behind one of the other two doors. The host's action concentrates that ²/₃ probability onto the remaining unopened door. Simulations and formal conditional probability both confirm this counter-intuitive result.
The weak law of large numbers states that the sample mean converges in probability to the true mean as n → ∞. In practice: the relative frequency of a fair coin landing heads approaches 0.5 as you flip more times, even if short runs deviate significantly. This does not mean a "due" event becomes more likely — each flip is independent (the gambler's fallacy).
Sensitivity (true positive rate), specificity (true negative rate), and prevalence together determine predictive values via Bayes' theorem. A test with 95% sensitivity and 95% specificity applied to a disease with 1% prevalence gives a positive predictive value of only ~16% — most positive results are false positives. This is why screening tests in low-prevalence populations require confirmatory testing before diagnosis.