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Probability Modeling and Conditioning Practice

Allow 60–90 minutes. For every answer state the experiment, outcomes, event, and assumptions. Use worked examples for A1–A2, then close them for A3–A5. Check the separate solutions after submitting an attempt.

  1. A1 — protocols. Flip three independent fair coins. Under protocol I, retain trials with at least one head. Under protocol II, choose one flip uniformly and retain the trial when that selected flip is head. For each protocol find the conditional probability of HHH and explain the weighting.
  2. A2 — detector. Among10,000 requests the defect prevalence is .02; sensitivity=.8, false-positive rate=.04. Build an expected count table, compute precision and recall, and calculate expected review volume.
  3. A3 — overlap. P(A)=.3, P(B)=.4, P(A intersect B)=.2. Find the union, P(A given B), and whether A and B are independent. Explain why adding .3 and .4 is wrong.
  4. A4 — changing evidence. A hypothesis has prior .1. Evidence E1 has likelihood .8 under H and .2 under not-H. E2 has likelihood .7 under H and .1 under not-H. Assuming E1,E2 conditionally independent under each hypothesis, compute the posterior after both. Then explain why the calculation is unjustified if E2 is a duplicate record of E1.
  5. A5 — bounds. Two events each have probability .9. Derive the smallest possible conditional probability of A given B. Construct an event table achieving the bound.

Repair errors by drawing a count table or probability tree. A correct number with an unstated observation protocol is incomplete. Transfer: choose a new prevalence and explain which cells change before recalculating.