Audit joint information and an adaptive reward rule
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 100 minutes, as an unvalidated planning estimate. This collection contains 2 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 joint law and information available at each decision.
Check the exact independence hypothesis rather than only marginal distributions.
Prove the claim for every allowed strategy before comparing selected examples.
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0 of 2 tasks have recorded responses. No overall mastery score is assigned.
Let R and S be independent uniform residues in {0,1,2,3,4}. Set T equal to 2S−R modulo 5. Prove that every pair among R,S,T is independent and uniform. For A={R=S} and B={S=T}, compute P(A), P(B) and P(A∩B). Does pairwise independence of the variables imply independence of these equality events? Now draw a fresh uniform R* independently of the entire retained pair (S,T). Compute P(R*=S and S=T), and identify the hypothesis that changed.
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.
Six distinct tokens are uniformly shuffled. Two carry value −2, three value 1, and one value 7. Before seeing the next token, choose either to take it and finish or to skip it and reveal its value. The last token must be taken. Decisions may depend on the revealed history and independent randomization. Prove that every such strategy gives each value the same selection probability as taking immediately, and compute the expected selected value. Next charge 1/4 reward unit for each skipped token. Analyze the policy that skips until the value 7 has been revealed, then takes the next token, except that the last token must always be taken. Find its expected skipped count and net reward. Finally change the information rule: the current token may be inspected before choosing whether to take or skip it. Under this different rule, compute the net reward of waiting for the value 7. Explain why it does not contradict the original strategy bound.
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
Let R and S be independent uniform residues in {0,1,2,3,4}. Set T equal to 2S−R modulo 5. Prove that every pair among R,S,T is independent and uniform. For A={R=S} and B={S=T}, compute P(A), P(B) and P(A∩B). Does pairwise independence of the variables imply independence of these equality events? Now draw a fresh uniform R* independently of the entire retained pair (S,T). Compute P(R*=S and S=T), and identify the hypothesis that changed.
Six distinct tokens are uniformly shuffled. Two carry value −2, three value 1, and one value 7. Before seeing the next token, choose either to take it and finish or to skip it and reveal its value. The last token must be taken. Decisions may depend on the revealed history and independent randomization. Prove that every such strategy gives each value the same selection probability as taking immediately, and compute the expected selected value. Next charge 1/4 reward unit for each skipped token. Analyze the policy that skips until the value 7 has been revealed, then takes the next token, except that the last token must always be taken. Find its expected skipped count and net reward. Finally change the information rule: the current token may be inspected before choosing whether to take or skip it. Under this different rule, compute the net reward of waiting for the value 7. Explain why it does not contradict the original strategy bound.
