Four manufactured switches are tested, and the recorded ordered outcomes are 1,0,1,1, where one means the switch passes. Before estimating a pass rate, write what was randomized: were four switches independently sampled from a continuing production process, or drawn without replacement from a small box? These produce different joint laws even if the marginal pass probability is the same. A statistical calculation starts with that distinction. The example is synthetic; its assumptions describe an imagined experiment, not evidence about an actual factory.
Predict
Under independent, identically distributed Bernoulli sampling, predict whether 1,0,1,1 and 1,1,1,0 have equal probabilities. Let p denote the unknown population pass probability. Independence multiplies probabilities, giving p³(1−p) for either vector. This equality is a model implication that could fail with temporal dependence. It does not mean every total is equally probable: there are four ordered vectors with three successes, but only one with four. Distinguish the probability of one recorded arrangement from the probability of its total.
Experiment
Enumerate the possible totals and count their arrangements. Check that the counts sum to sixteen, the number of binary vectors of length four. Then set p=1/2 and divide each count by sixteen to obtain the distribution of the total. Explain why setting p=1/4 changes the probabilities without changing the sample space.
Total successes s
Number of arrangements
Probability of the total
0
1
(1−p)⁴
1
4
4p(1−p)³
2
6
6p²(1−p)²
3
4
4p³(1−p)
4
1
p⁴
Understand
A statistical model is a family of probability laws indexed by a parameter θ. The sample space lists possible observations; the parameter space lists candidate population descriptions. A dominated model has densities fθ relative to one common reference measure, such as counting measure for finite outcomes or Lebesgue measure for continuous measurements. A density is not necessarily a point probability. This distinction becomes essential when likelihoods are introduced.
A statistic is a function of the observed data that does not require an unknown parameter to evaluate. The total S=ΣXᵢ is a statistic; so is the sample mean. In contrast, Σ(Xᵢ−μ)² with unknown μ is a parameter-dependent random quantity, not a statistic. Such quantities can still be useful in theoretical pivots. “Not a statistic” does not mean “mathematically forbidden”; it tells us what can actually be computed from the data alone.
Apply
Now suppose the box contains exactly three passing switches and one failing switch, and two are selected without replacement. Each first draw passes with probability 3/4. After a pass, the next conditional pass probability is 2/3; after a failure it is 1. Multiplying 3/4 by 3/4 would incorrectly assume independence. Compute the probability of two passes as (3/4)(2/3)=1/2 and compare it with the iid value 9/16. Write a sentence identifying which physical sampling operation changes the calculation.
Also challenge a convenient shortcut: a large sample does not establish that the observations represent the target population. A selection process can consistently sample the wrong population. Statistical precision under a model and credibility of the model are separate questions.
Continue
For every later derivation, keep a short model ledger: what is observed, what is fixed, what is unknown, how observations depend on each other, and what population the claim concerns. Add units when observations are measured quantities. These entries determine which theorems can be used. Review Probability for joint laws and conditioning. Continue to Likelihood and identifiability, where the observed vector is held fixed while candidate parameter values vary.
Apply · explain · revise
Practice and reasoning
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Four ordered binary observations have how many possible sample vectors?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For four iid Bernoulli(p) observations, how many vectors have exactly two successes?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Write the joint probability of a binary vector x of length n under iid Bernoulli(p), and identify its reference measure.
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.
Explain why drawing two items without replacement from a finite binary population is not an iid Bernoulli sample.
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.
For normal data with unknown μ, decide whether Σ(Xᵢ−μ)² is a statistic and repair it if necessary.
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.
A large convenience sample is analyzed as iid draws from all adults. Identify a mathematical assumption that sample size cannot verify.
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.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
