Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 120 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Define the random experiment and observed event before conditioning.
State independence, restart and test-cost assumptions before deriving a formula.
Keep exact probabilities, approximations, finite-cap outcomes and model expectations distinct.
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0 of 3 tasks have recorded responses. No overall mastery score is assigned.
A uniform three-token hand is drawn from eight distinct tokens, exactly three marked. Let J count marked tokens. Compare three reports: C reports whenever the hand contains a marked token; R samples one uniformly random hand member and reports only if that token is marked; S reports with probability j/4 when J=j. Derive the distribution of J. For each protocol compute both the report probability and the conditional probability that J is at least 2 given a report. Can equal conditional answers establish that R and S are the same reporting mechanism?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Each fresh attempt tosses two independent coins with head probability1/3. HH is success, TT is failure, and either mixed pair restarts. Attempts use new independent tosses. Compute eventual success probability, expected attempts and expected restarts. If at most three attempts are allowed, separately compute success, failure and unresolved probabilities and expected attempts. Now consider a different process: choose a hidden mode once, each with probability1/2. In mode G every attempt independently succeeds with probability1/2 or restarts with probability1/2; in mode B the first attempt always fails. Show why applying the fresh-restart ratio to unconditional first-attempt probabilities gives the wrong eventual success answer.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A batch contains exactly one defective item among twelve, uniformly located. Partition it into three fixed pools of four. Each pool test costs one and detects a defect perfectly; every positive pool is followed by all four individual tests. Compute the total tests in this fixed-count model. Compare it with the expected total in an independent-item model where each of twelve items is defective with probability1/12. Then use one three-level binary inspection tree on a group of eight independently defective items with probability p: test the root group; after a positive internal group, test both equal children, including singleton children. Derive the expected test count, counting both children even if one positive result would logically determine the other. Check p=0 and p=1 and explain why this is not an optimized inference-aware strategy.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
A uniform three-token hand is drawn from eight distinct tokens, exactly three marked. Let J count marked tokens. Compare three reports: C reports whenever the hand contains a marked token; R samples one uniformly random hand member and reports only if that token is marked; S reports with probability j/4 when J=j. Derive the distribution of J. For each protocol compute both the report probability and the conditional probability that J is at least 2 given a report. Can equal conditional answers establish that R and S are the same reporting mechanism?
Each fresh attempt tosses two independent coins with head probability1/3. HH is success, TT is failure, and either mixed pair restarts. Attempts use new independent tosses. Compute eventual success probability, expected attempts and expected restarts. If at most three attempts are allowed, separately compute success, failure and unresolved probabilities and expected attempts. Now consider a different process: choose a hidden mode once, each with probability1/2. In mode G every attempt independently succeeds with probability1/2 or restarts with probability1/2; in mode B the first attempt always fails. Show why applying the fresh-restart ratio to unconditional first-attempt probabilities gives the wrong eventual success answer.
A batch contains exactly one defective item among twelve, uniformly located. Partition it into three fixed pools of four. Each pool test costs one and detects a defect perfectly; every positive pool is followed by all four individual tests. Compute the total tests in this fixed-count model. Compare it with the expected total in an independent-item model where each of twelve items is defective with probability1/12. Then use one three-level binary inspection tree on a group of eight independently defective items with probability p: test the root group; after a positive internal group, test both equal children, including singleton children. Derive the expected test count, counting both children even if one positive result would logically determine the other. Check p=0 and p=1 and explain why this is not an optimized inference-aware strategy.
