Probability foundations: count the experiment you actually observed
“Some amber item was drawn” and “a randomly revealed item was amber” are different observations. Build the sample space and the reporting mechanism before applying a formula. Choose a probability spine or a statistics alternative according to the actual course question.
Course topic: Probability and Statistics
Study route and exercise: University Reading List editorial project. How we curate · Sources checked:
Scope and responsibility
A first route through finite probability models, conditional events and expectation within the broader probability-statistics course topic. It is not a full statistical-inference course. Grinstead–Snell is the author-hosted complete text, Blitzstein–Hwang the 2019 second edition, and OpenStax the 2023 second edition. All examples below are invented sampling experiments with stipulated randomization; no empirical defect rate, product performance or population finding is claimed.
Choose your starting point
I want a free probability spine
Use Grinstead–Snell for discrete and continuous models, simulations and later limit theorems. Begin with the discrete route rather than assuming that “free” also means calculus-free.
I want conditional reasoning and distribution stories
Choose Blitzstein–Hwang when its problem-solving approach suits you. Harvard links an authorized free second edition; use that entry or a legitimate copy and verify the edition.
My course asks about data and inference
Use OpenStax for sampling, descriptive statistics and inference. Its probability chapters are supporting preparation, not a replacement for a full calculus-based probability text.
Compare the three books
Introduction to Probability
Charles M. Grinstead; J. Laurie Snell · Edition: Complete author-hosted textbook
Free probability course with simulation
- Choose it when
- You want a sustained probability route and can support its later continuous material with calculus.
- Use another route when
- You need an algebra-only statistics survey or only current software instructions.
- Selected reading
- Chapters 1, 3, 4 and 6 for discrete models, counting, conditioning and expectation; add Chapters 2 and 5 for continuous distributions later.
- Limits and trade-offs
- The official page supplies the full book and odd-numbered exercise answers. A simulation can suggest a check, but its frequencies do not prove an exact probability.
Try before choosing
List the ten equally likely unordered pairs in the five-token example, then mark the conditioning event before dividing.
Check your work: Three pairs are AA, six mixed and one BB. The conditioning denominator is nine pairs, not five tokens or ten unconditioned pairs.
Introduction to Probability
Joseph K. Blitzstein; Jessica Hwang · Edition: 2 · 2019
Conditional problem-solving alternative
- Choose it when
- You want to justify distributions through situations and use more than formula substitution.
- Use another route when
- You mainly need confidence intervals and hypothesis tests, or assume a book title establishes identical chapter numbering.
- Selected reading
- Second-edition opening chapters: Probability and Counting; Conditional Probability; Random Variables; Expectation.
- Limits and trade-offs
- The selected ISBN is the 2019 hardback; ebook and course tools have other identifiers. Harvard’s free-book link establishes authorized reading, not unrestricted reuse or paid course assessment.
Try before choosing
Solve the reveal-protocol follow-up by conditioning on pair type, then re-explain it by considering the remaining unseen token.
Check your work: The two arguments give 1/2 only under the stated uniform reveal rule. An observer who always chooses an amber token generates a different likelihood.
Introductory Statistics 2e
Barbara Illowsky; Susan Dean · Edition: 2 · 2023
Free algebra-based statistics alternative
- Choose it when
- You need the route from collected data to estimates, tests and regression rather than probability theory alone.
- Use another route when
- You expect it to cover the full calculus-based probability development or justify an inference from an invented probability model.
- Selected reading
- Chapters 3–5 for supporting probability ideas; 1–2 for data, then 8–9 for an introduction to estimation and testing.
- Limits and trade-offs
- Its preface assumes intermediate algebra and emphasizes applications. Free web/PDF access and the current BY-NC-SA licence are separate from permission to reuse figures commercially.
Try before choosing
Write two different questions: a probability under a fully stated draw model, and an estimate about an unknown real population. List the extra evidence the second needs.
Check your work: The model answer alone supplies no observed sample or defensible sampling design. A textbook exercise is not a confidence interval for an actual population.
Check your preparation
What is the outcome unit?
Specify whether outcomes are ordered draws or unordered pairs. Either can work, but numerator and denominator must use the same unit and the correct probabilities.
Does drawing without replacement preserve independence?
No in general. After one amber token, only two amber tokens remain among four. Equal marginal probabilities for positions do not imply independent positions.
Does expectation require independence?
Linearity does not. Two draw indicators can be dependent while their expectations still add. Independence is a separate condition used in other formulas.
Six steps with a concrete output
Outputs marked ready: 0 / 6
These are your own study marks, not an assessment. They are not sent to this site.
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Define the experiment
Selected reading: Grinstead–Snell 1/3; Blitzstein–Hwang Probability and Counting.
Make this: Specify tokens, replacement, ordering and equal-likelihood assumptions in a short model card.
Check your work: The ten unordered pairs are equally likely here because sampling is uniform without replacement; do not assume that property for every description.
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Count the event
Selected reading: The same counting sections.
Make this: Enumerate AA, mixed and BB pairs, and check that their counts exhaust the sample space.
Check your work: 3+6+1=10. Do not double-count unordered mixed pairs or mix them with ordered AA counts.
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Condition explicitly
Selected reading: Grinstead–Snell 4; Blitzstein–Hwang Conditional Probability.
Make this: Solve the main problem using an intersection and the event probability.
Check your work: The conditioning event has positive probability. Explain which outcomes are removed rather than merely naming Bayes’ rule.
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Use indicators
Selected reading: Grinstead–Snell 6 or Blitzstein–Hwang Expectation.
Make this: Calculate the expected number of amber tokens in two draws and explain why dependence does not break this argument.
Check your work: E[X]=2×3/5=6/5. This is an average over repeated model experiments, not a possible fractional count in a single pair.
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Change the reporting mechanism
Selected reading: Return to the conditional-probability sections.
Make this: Draw a tree for the follow-up, including the clerk’s random reveal after the pair is drawn.
Check your work: Separate the draw probability from the probability of the message given a pair. Do not discard the second random stage.
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Separate model from inference
Selected reading: OpenStax 1/3 and, if needed,8–9.
Make this: Write an exact model result and a paragraph stating what an actual data claim would additionally require.
Check your work: Keep known token composition separate from estimating an unknown composition. Do not claim observations that this invented exercise never collected.
Worked exercise
Original example: condition on at least one amber token
A box contains three distinct amber tokens and two distinct blue tokens. Draw two uniformly without replacement. Given that the pair contains at least one amber token, find the probability both are amber. Also find the expected amber count before the information is supplied.
Assumptions
All five tokens are distinct despite sharing colours. Every unordered pair is equally likely. “At least one amber” is a truthful event report without any further selection stage. All objects and observations are invented.
- There are C(5,2)=10 pairs: C(3,2)=3 AA,3×2=6 mixed and C(2,2)=1 BB. The event at least one amber contains 9 pairs.
- P(AA | at least one amber)=(3/10)/(9/10)=1/3. Equivalently,3 of the 9 surviving equally likely pairs are AA.
- Let X=I₁+I₂ count amber tokens. Each position has amber probability 3/5, so E[X]=6/5 even though the indicators are dependent.
Answer and interpretation
The conditional probability is 1/3; the unconditional expected amber count is 6/5. Multiplying (3/5)² would incorrectly assume replacement or independence and give 9/25 instead of the unconditional AA probability 3/10.
Test the result
Use ordered draws as an independent count: there are 20 ordered outcomes,6 AA and 18 with at least one amber;6/18=1/3. The outcome unit changes but the final probability does not.
Change an assumption
Change the message, not the draw
Keep the same box and uniform two-token draw. Replace the event report: a clerk uniformly chooses one of the two drawn tokens, independently of their colours given the pair, and reveals it. You see amber. Find the probability that both drawn tokens are amber. Then compare a clerk who always shows amber whenever possible.
- Before revealing, pair probabilities are AA 3/10, mixed 6/10 and BB 1/10. Under uniform reveal, amber is shown with probabilities 1, 1/2 and 0 respectively.
- P(shown amber)=3/10+(6/10)(1/2)=3/5. Therefore P(AA | shown amber)=(3/10)/(3/5)=1/2.
- If the clerk always selects amber when available, shown amber is exactly the old at-least-one event and the result returns to 1/3. The underlying draw has not changed.
Answer and interpretation
A uniform reveal gives 1/2; an amber-preferring reveal gives 1/3. The information mechanism is part of the model, so the phrase “I saw amber” is not enough by itself to select the denominator.
Test the result
After a uniformly revealed amber token, the unseen token is uniform among the other four original tokens: two amber and two blue. This gives 1/2 again. That shortcut depends on the stated reveal protocol, not on a general rule that all hidden possibilities are equally likely.
Common mistakes
Conditioning is not replacement
Removing outcomes after information is different from physically returning a token before the second draw.
Simulation is not an exact proof
Random frequencies have sampling variation; use enumeration or algebra for the exact result here.
Probability is not automatically inference
The model stipulates composition and randomization. Actual estimates require data and a defensible relation to a target population.
Before moving on
- State the sample space and observation protocol separately.
- Derive both 1/3 and 1/2 with their different conditioning events.
- Explain which use of expectation does not require independence.
Sources and edition checks
Sources checked:
- Dartmouth: complete text, chapter map and selected answers
The official map confirms discrete/continuous scope, named chapters and odd-answer access.
- CRC: selected second-edition identity and contents
The publisher confirms second-edition 2019 hardback identity and the selected opening topics.
- Harvard Stat110: author-authorized free second edition
Harvard explicitly links a free online second edition through probabilitybook.net; this is an authorized access entry rather than a promise that every automated PDF fetch works.
- OpenStax: statistics scope, preparation and contents
The preface establishes the algebra-based statistics audience, probability/inference split and current access/licence terms.
These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.