Probability Fundamentals
Events, conditional probability, and Bayes' theorem — the building blocks every statistical method assumes.
Try answering in your head first, then click a question to check the model answer.
Q1.Explain the difference between independent events and mutually exclusive events.(show answer)
Independent events don't influence each other's probability — knowing one occurred tells you nothing about the other. Mutually exclusive events can't occur together at all (P(A and B) = 0) — which actually means they're strongly dependent: knowing one occurred tells you the other definitely didn't. These are often confused but are close to opposite concepts.
Q2.State Bayes' theorem and explain what each term means in a concrete example.(show answer)
P(A|B) = P(B|A) × P(A) / P(B). Example: for a medical test, P(disease|positive test) — what we want — is computed from P(positive test|disease) (the test's sensitivity), P(disease) (the prior/base rate), and P(positive test) (overall probability of a positive result). This is how you correctly update a belief given new evidence, rather than confusing P(A|B) with P(B|A).
Q3.Why is base rate neglect a common statistical mistake, and how does Bayes' theorem correct it?(show answer)
People often estimate P(disease|positive test) as if it equals the test's accuracy (P(positive|disease)), ignoring how rare the disease actually is (the prior). For a rare disease, even an accurate test produces many false positives relative to true positives, so P(disease|positive test) can be surprisingly low. Bayes' theorem forces you to explicitly incorporate the prior P(disease), correcting this intuition.
Q4.What's the difference between expected value and variance, and why do you need both to describe a distribution?(show answer)
Expected value is the long-run average outcome — where the distribution is centered. Variance measures how spread out outcomes are around that average. Two distributions can have the identical expected value but very different variance (e.g. a safe bet vs. a high-risk/high-reward bet with the same average payout) — the average alone doesn't tell you about risk or predictability.