intro practice · coding problem
Independence and inclusion-exclusion
P(A)=0.4, P(B)=0.5, P(A∩B)=0.2. Are A and B independent? Compute P(A∪B). If they were mutually exclusive, what would P(A∪B) be instead?
The mechanism
Probability is a bookkeeping system for uncertainty. Conditional probability restricts the sample space; Bayes flips cause and effect — from 'how likely is the evidence given the hypothesis' to 'how likely is the hypothesis given the evidence'.
Expected evidence
P(A)P(B)=0.2=P(A∩B) → independent. P(A∪B)=0.4+0.5−0.2=0.7. Mutually exclusive would give 0.9 — conflating the two is a common bug.
Try this drill in the practice workspace →
Hints when you need them
- Independent ⟺ P(A∩B)=P(A)P(B).
- Union: P(A)+P(B)−P(A∩B).
Watch for these mistakes
- Treating independent events as mutually exclusive
- Confusing P(A|B) with P(B|A) (the prosecutor's fallacy)
- Adding probabilities of overlapping events without inclusion-exclusion