Connect arrivals, route outcomes and incomplete evidence
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 150 minutes, as an unvalidated planning estimate. This collection contains 4 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.
Attempt the four original mixed tasks before opening their worked reasoning. Record assistance and solution exposure.
Derive the relevant joint law or termination condition; a matching decimal alone is insufficient.
Keep physical time, route transitions, finite cap, numerical approximation and model assumptions separate. This is public practice, not a secure independent examination.
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0 of 4 tasks have recorded responses. No overall mastery score is assigned.
A rate-2-per-minute exponential-wait construction is observed for 3/2 minutes. Independently label each arrival D with probability 1/3. Derive the probability of exactly two D and one C arrivals, then find the probability that the second D arrival has occurred by 3/2 minutes. Give two derivations for the second result and state which assumptions make the D process Poisson.
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.
Take Q=[[1/2,1/4],[0,1/2]] and r_D=(0,1/4)ᵀ, with remaining mass going to absorbing C. Starting at A, derive eventual D probability, expected transitions, D-by-two-steps probability and unresolved probability at two steps. Supply a proof that the eventual calculations are valid and explain why a cap is not a new absorbing outcome.
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 one-hour count has mean and variance 2 under two proposed mechanisms: independent successive Poisson(2) counts, or one Poisson(2) draw copied into the next window. Predict the means, pair covariance, total variance and pair-zero frequency. Then design a paired-record check with a stated finite-sample error bound and explain why matching both marginal mean and variance does not settle the choice.
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 new chain has A→B,C each with probability 1/2, B→B surely and absorbing C,D. An analyst runs 128 paths from A, discards paths still at B at a cap, and reports certain absorption at C with a mean of one transition. Diagnose the error using exact probabilities. Propose a report that retains the actual model conclusion, the finite observed evidence and a separate route for repairing the model if the trapped state is unrealistic.
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 rate-2-per-minute exponential-wait construction is observed for 3/2 minutes. Independently label each arrival D with probability 1/3. Derive the probability of exactly two D and one C arrivals, then find the probability that the second D arrival has occurred by 3/2 minutes. Give two derivations for the second result and state which assumptions make the D process Poisson.
Take Q=[[1/2,1/4],[0,1/2]] and r_D=(0,1/4)ᵀ, with remaining mass going to absorbing C. Starting at A, derive eventual D probability, expected transitions, D-by-two-steps probability and unresolved probability at two steps. Supply a proof that the eventual calculations are valid and explain why a cap is not a new absorbing outcome.
A one-hour count has mean and variance 2 under two proposed mechanisms: independent successive Poisson(2) counts, or one Poisson(2) draw copied into the next window. Predict the means, pair covariance, total variance and pair-zero frequency. Then design a paired-record check with a stated finite-sample error bound and explain why matching both marginal mean and variance does not settle the choice.
A new chain has A→B,C each with probability 1/2, B→B surely and absorbing C,D. An analyst runs 128 paths from A, discards paths still at B at a cap, and reports certain absorption at C with a mean of one transition. Diagnose the error using exact probabilities. Propose a report that retains the actual model conclusion, the finite observed evidence and a separate route for repairing the model if the trapped state is unrealistic.
