Probability: laws, dependence and justified limits
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 210 minutes, as an unvalidated planning estimate. This collection contains 14 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.
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A uniform three-object hand is drawn from eight objects, three marked. Compute the conditional probability that all three hand members are marked after(i) being told at least one is present,(ii) a uniformly selected member being marked,(iii) a host deliberately selecting a marked member whenever possible and reporting only its marked status. Give the two distinct conditional means. If the uniform reveal identifies a particular marked object, justify the remaining-hand law rather than assuming it.
Structure your argument and preserve revisions
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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.
Three indicators have success probabilities1/5,2/5,4/5. For independent indicators compute the mean, variance and exact probability their total reaches3, then compute Markov and Cantelli upper bounds. Replace independence with a shared U uniform on[0,1], each indicator testing U<p_i. Derive the entire count law and correct variance/bound. Does the former Cantelli value remain a guarantee? Finally give a variance bound requiring only the known support and mean.
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.
Let independent Bernoulli(1/2) trials form disjoint blocks of lengths1,2,4. Let X count all-success blocks. Find E[X], Var(X) and P(X≥1), justifying indicator independence. Now reuse prefixes of lengths1,2,4 of a single four-trial sequence, and call the count Y. Derive its entire law, mean and variance. Explain why neither the old probability of at least one success nor variance addition can be carried over.
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.
Protocol A repeats iid Bernoulli(1/4) trials until three successes, but records only through trial4. Derive the completed stopping masses, censored probability, full stopping mean/variance and mean among completed cases. Explain which would change if one coin outcome were reused forever. Protocol B records four iid category labels with p=(1/4,1/2,1/4); protocol C instead copies each of two iid labels twice. Derive B’s probability of counts(1,2,1), its conditional law of N_A given N_C=1, and Cov(N_A,N_B). Contrast C’s support and covariance. Explain why neither a finite cap nor equal mean counts licenses the simpler iid interpretation.
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 law assigns probability 1/4 to X=0 and otherwise draws uniformly on (0,4). Give its CDF on the whole real line. Compute P(0≤X≤2) and P(0<X≤2). For Aₙ={0≤X<1/n} and Bₙ={0<X<1/n}, identify the intersections and limits of their probabilities, citing the applicable axiom-derived result.
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.
Let U,V be independent and uniform on {0,1,2}, and let W be their sum modulo 3. Show the three variables are pairwise independent but not mutually independent. Separately, source A has prior probability 1/5 and produces a signal with probability 3/4; source B has prior 4/5 and signal probability 1/8. Find the signal probability and posterior source probability, making the partition assumptions explicit.
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.
Independent Bernoulli trials succeed with probability 1/5. Find the probability of exactly two successes in five trials, the chance the first success occurs after trial 3, and the mean trial index of the first success. Separately, independent Poisson counts have means 2 and 3; derive the distribution of their sum. Explain why neither a geometric nor Poisson model is justified solely by knowing a mean.
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.
Normalize f(x)=cx² on 0<x<3 and zero elsewhere. Compute its CDF, mean, variance and E[(X−1)²]. Recover the mean independently from a survival integral, and explain why a density height greater than 1 is allowed.
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.
Let X be uniform on(−3,1) and Y=X². Derive the full CDF and density of Y, separating ranges where one or two inverse branches contribute. Check normalization and identify whether Y has an atom at 0 or 1.
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 joint density is 3x on 0<y<x<1. Verify normalization, derive both marginals, and find the conditional law of Y given X=x for 0<x<1. Compute E(X), E(Y) and E(XY). Is the pair independent? Explain the meaning of density conditioning at a point.
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.
In the preceding triangular model, use E(X²)=3/5 and Y|X=x uniform on(0,x) to derive Var(Y) by total variance. Find Var(2Y−X). Prove that predicting Y by X/2 minimizes mean squared error among square-integrable functions of X; explain why zero conditional mean of 2Y−X does not imply independence from X.
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.
Let U,V be independent exponential variables with respective rates 2 and 3. Derive the density of S=U+V by convolution and its MGF with domain. Then set R=U/(U+V). Use the inverse transformation to derive the joint density of (S,R), find the marginal density of R, and decide whether S and R are independent.
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.
Compare three separate models: U uniform on(1,5), T exponential with rate 0.5 per hour, and Z normal with mean 12 and SD 3. Find P(U>3), P(T>6 hours|T>2 hours), and P(9≤Z≤18), using Φ(1)=0.841345 and Φ(2)=0.977250. State which result uses memorylessness and which uses symmetry or standardization.
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.
Independent identically distributed observations have mean 2 and variance 16. Use Chebyshev to find a sufficient n ensuring P(|X̄−2|≥0.5)≤0.01. At n=64, approximate P(1.5≤X̄≤2.5) by the classical CLT using Φ(1)=0.841345. Explain why this approximation has no guaranteed accuracy from those moments alone. Also derive E(X²)≥[E(X)]² through either Jensen or nonnegative variance.
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-object hand is drawn from eight objects, three marked. Compute the conditional probability that all three hand members are marked after(i) being told at least one is present,(ii) a uniformly selected member being marked,(iii) a host deliberately selecting a marked member whenever possible and reporting only its marked status. Give the two distinct conditional means. If the uniform reveal identifies a particular marked object, justify the remaining-hand law rather than assuming it.
Three indicators have success probabilities1/5,2/5,4/5. For independent indicators compute the mean, variance and exact probability their total reaches3, then compute Markov and Cantelli upper bounds. Replace independence with a shared U uniform on[0,1], each indicator testing U<p_i. Derive the entire count law and correct variance/bound. Does the former Cantelli value remain a guarantee? Finally give a variance bound requiring only the known support and mean.
Let independent Bernoulli(1/2) trials form disjoint blocks of lengths1,2,4. Let X count all-success blocks. Find E[X], Var(X) and P(X≥1), justifying indicator independence. Now reuse prefixes of lengths1,2,4 of a single four-trial sequence, and call the count Y. Derive its entire law, mean and variance. Explain why neither the old probability of at least one success nor variance addition can be carried over.
Protocol A repeats iid Bernoulli(1/4) trials until three successes, but records only through trial4. Derive the completed stopping masses, censored probability, full stopping mean/variance and mean among completed cases. Explain which would change if one coin outcome were reused forever. Protocol B records four iid category labels with p=(1/4,1/2,1/4); protocol C instead copies each of two iid labels twice. Derive B’s probability of counts(1,2,1), its conditional law of N_A given N_C=1, and Cov(N_A,N_B). Contrast C’s support and covariance. Explain why neither a finite cap nor equal mean counts licenses the simpler iid interpretation.
A law assigns probability 1/4 to X=0 and otherwise draws uniformly on (0,4). Give its CDF on the whole real line. Compute P(0≤X≤2) and P(0<X≤2). For Aₙ={0≤X<1/n} and Bₙ={0<X<1/n}, identify the intersections and limits of their probabilities, citing the applicable axiom-derived result.
Let U,V be independent and uniform on {0,1,2}, and let W be their sum modulo 3. Show the three variables are pairwise independent but not mutually independent. Separately, source A has prior probability 1/5 and produces a signal with probability 3/4; source B has prior 4/5 and signal probability 1/8. Find the signal probability and posterior source probability, making the partition assumptions explicit.
Independent Bernoulli trials succeed with probability 1/5. Find the probability of exactly two successes in five trials, the chance the first success occurs after trial 3, and the mean trial index of the first success. Separately, independent Poisson counts have means 2 and 3; derive the distribution of their sum. Explain why neither a geometric nor Poisson model is justified solely by knowing a mean.
Normalize f(x)=cx² on 0<x<3 and zero elsewhere. Compute its CDF, mean, variance and E[(X−1)²]. Recover the mean independently from a survival integral, and explain why a density height greater than 1 is allowed.
Let X be uniform on(−3,1) and Y=X². Derive the full CDF and density of Y, separating ranges where one or two inverse branches contribute. Check normalization and identify whether Y has an atom at 0 or 1.
A joint density is 3x on 0<y<x<1. Verify normalization, derive both marginals, and find the conditional law of Y given X=x for 0<x<1. Compute E(X), E(Y) and E(XY). Is the pair independent? Explain the meaning of density conditioning at a point.
In the preceding triangular model, use E(X²)=3/5 and Y|X=x uniform on(0,x) to derive Var(Y) by total variance. Find Var(2Y−X). Prove that predicting Y by X/2 minimizes mean squared error among square-integrable functions of X; explain why zero conditional mean of 2Y−X does not imply independence from X.
Let U,V be independent exponential variables with respective rates 2 and 3. Derive the density of S=U+V by convolution and its MGF with domain. Then set R=U/(U+V). Use the inverse transformation to derive the joint density of (S,R), find the marginal density of R, and decide whether S and R are independent.
Compare three separate models: U uniform on(1,5), T exponential with rate 0.5 per hour, and Z normal with mean 12 and SD 3. Find P(U>3), P(T>6 hours|T>2 hours), and P(9≤Z≤18), using Φ(1)=0.841345 and Φ(2)=0.977250. State which result uses memorylessness and which uses symmetry or standardization.
Independent identically distributed observations have mean 2 and variance 16. Use Chebyshev to find a sufficient n ensuring P(|X̄−2|≥0.5)≤0.01. At n=64, approximate P(1.5≤X̄≤2.5) by the classical CLT using Φ(1)=0.841345. Explain why this approximation has no guaranteed accuracy from those moments alone. Also derive E(X²)≥[E(X)]² through either Jensen or nonnegative variance.
