Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 90 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.
State whether draws are independent, uniform and with replacement. Marginal probabilities alone do not determine a joint law.
Separate expected counts, event probabilities and deterministic guarantees. Explain the direction of every bound.
Try these changed cases before reading the public solutions; written arguments require review.
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A sampler has four labels, numbered 0 through 3. Mechanism A makes four iid uniform draws with replacement. Mechanism B chooses one uniform starting label U, then emits U,U+1,U+2,U+3 modulo 4. Mechanism C chooses U uniformly, then emits U four times. Prove that every individual position is uniform under all three mechanisms. For each mechanism find the probability of complete collection, the probability of any collision and the expected number of matching position pairs. Explain why applying the iid collision product or coupon waiting-stage formula to B or C would be invalid. If B and C continue their respective cyclic or constant rules forever, find their full-collection times.
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 test process independently selects one of four equally likely categories on each trial. Let T be the number of trials until every category has appeared. Derive E[T]. For integer t≥1, use inclusion-exclusion to prove P(T≤t) = 1 − 4(3/4)^t + 6(1/2)^t − 4(1/4)^t. Find the actual smallest integer deadline giving at least95% completion, using the neighboring values at 15 and16 and a monotonicity argument. Compare this with the sufficient deadlines from 4(3/4)^t and 4exp(−t/4) bounding failure. Finally determine the probability of a collision by draw 5 and explain why this says very little about whether all categories have appeared.
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 sampler has four labels, numbered 0 through 3. Mechanism A makes four iid uniform draws with replacement. Mechanism B chooses one uniform starting label U, then emits U,U+1,U+2,U+3 modulo 4. Mechanism C chooses U uniformly, then emits U four times. Prove that every individual position is uniform under all three mechanisms. For each mechanism find the probability of complete collection, the probability of any collision and the expected number of matching position pairs. Explain why applying the iid collision product or coupon waiting-stage formula to B or C would be invalid. If B and C continue their respective cyclic or constant rules forever, find their full-collection times.
A test process independently selects one of four equally likely categories on each trial. Let T be the number of trials until every category has appeared. Derive E[T]. For integer t≥1, use inclusion-exclusion to prove P(T≤t) = 1 − 4(3/4)^t + 6(1/2)^t − 4(1/4)^t. Find the actual smallest integer deadline giving at least95% completion, using the neighboring values at 15 and16 and a monotonicity argument. Compare this with the sufficient deadlines from 4(3/4)^t and 4exp(−t/4) bounding failure. Finally determine the probability of a collision by draw 5 and explain why this says very little about whether all categories have appeared.
