piatra . institutepathwaysFinding the next step…
Connect probability models, conditioning, random variables and inference with simulation and explicit assumptions.
No subject-specific background required; work through the opening foundation.
A simulation and statistical analysis with an uncertainty statement and model critique.
Barbara Illowsky · Susan Dean · reference
Named section and worked examples
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Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
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Open the readingStatistics: terminology. Distinguish sample and population.
Statistics: data and sampling. Identify sampling bias.
Barbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingProbability: terminology. Define events and a sample space.
Probability: independent and mutually exclusive events. Explain why these are different properties.
Probability: two basic rules. Calculate unions and intersections.
Probability: contingency tables. Compute a conditional probability from counts.
Events A and B are mutually exclusive, with P(A) = 0.3 and P(B) = 0.4. Find P(A or B). Are they independent?
Compare the actual intersection with the product required by independence.
P(A or B) = 0.3 + 0.4 = 0.7 because the intersection is zero. They are not independent: independence would require P(A and B) = 0.3 × 0.4 = 0.12. In fact, learning that B occurred makes A impossible.
What to look for
Common mistake: Treating “cannot occur together” as the meaning of independence.
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Barbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingProbability: discrete distributions. Check normalisation.
Probability: expected value. Compare an average with a single outcome.
Probability: binomial distribution. State the independence and constant-probability assumptions.
Statistics: central limit theorem. Simulate the distribution of a mean.
A fair coin with independent flips has produced five heads in a row. What is the probability of heads on the next flip, given that history?
The question is about the next flip after the observed sequence, not the probability of predicting the whole sequence in advance.
The conditional probability is still 1/2. Under the stated independence assumption, earlier results do not change the next flip’s distribution. The probability of five specified heads before any flips is 1/32, but that is a different question.
What to look for
Common mistake: Expecting tails to become more likely because it is “due.”
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Barbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingBarbara Illowsky · Susan Dean · reference
Named section and worked examples
Check reading access
Open the readingStatistics: experimental design. Separate random assignment from random sampling.
Statistics: confidence intervals. State the repeated-sampling interpretation.
Statistics: hypothesis testing. Choose a testable claim before seeing the result.
In an invented survey, 80% of 100 volunteer app users say they liked a course. Can you conclude that 80% of everyone in the town would like it? What would improve the evidence?
Check who could enter the sample and who chose to respond.
No. The reported percentage describes the respondents. App access and voluntary participation can select a group that differs from the town’s population. Define the target population, use an appropriate sampling frame and a selection method that reaches its relevant groups, and report nonresponse and uncertainty. A larger sample of the same volunteers does not by itself remove selection bias.
What to look for
Common mistake: Treating sample size as a cure for an unrepresentative sampling method.
Start this pathway to keep your answers in My learning.
Editorial perspectives based on selected works, rather than author-endorsed reading lists.
An editorial comparison of the core argument, alternative explanations and practical evidence.
A curated synthesis cannot be neutral or exhaustive. Follow the primary sources when an interpretation matters.
An editorial route that starts with working examples.
The order supports a particular study method; it does not establish which account is true.
An editorial route that starts with competing explanations.
The order supports a particular study method; it does not establish which account is true.
Background, different viewpoints, and further reading.
Distinguish sample and population.
Named section and worked examples
Identify sampling bias.
Named section and worked examples
Separate random assignment from random sampling.
Named section and worked examples
Inspect distribution shape before calculating.
Named section and worked examples
Compare mean and median under outliers.
Named section and worked examples
Interpret variation in the original units.
Named section and worked examples
Define events and a sample space.
Named section and worked examples
Explain why these are different properties.
Named section and worked examples
Calculate unions and intersections.
Named section and worked examples
Compute a conditional probability from counts.
Named section and worked examples
Trace a sequence of conditional events.
Named section and worked examples
Check normalisation.
Named section and worked examples
Compare an average with a single outcome.
Named section and worked examples
State the independence and constant-probability assumptions.
Named section and worked examples
Use density to calculate interval probabilities.
Named section and worked examples
Inspect the memoryless property.
Named section and worked examples
Standardise a measurement.
Named section and worked examples
Simulate the distribution of a mean.
Named section and worked examples
State the repeated-sampling interpretation.
Named section and worked examples
Choose a testable claim before seeing the result.
Named section and worked examples
Use probability for the optional confidence-interval and hypothesis-test readings; the first data report needs only arithmetic.
Use probability and uncertainty to examine model outputs and held-out evaluation.
Use conditional probability and careful event definitions in the worked calculations.
Practise conditional probability before comparing generative and predictive accounts.
Use probability and state counting for the entropy task; review algebra and logarithms too.
Use probability for the later causal-inference examples; directed diagrams are enough to begin.
Use conditional probability to separate an observed association from an intervention question.
Bayes’ rule and probability distributions support the selected mathematical reading; equivalent experience is welcome.
The full book uses probabilistic inference throughout its models; this route supplies a useful starting foundation.
Probability supports the paper’s treatment of uncertainty and information; logarithms are also helpful.