Mistake Clinic
Use counterexamples to expose the model error before repairing arithmetic. Keep real failed attempts; there is no requirement to manufacture a quota of mistakes.
Module-Specific Mistake Radar
| Wrong claim | Small counterexample or diagnostic | Repair evidence |
|---|---|---|
| Three possible head counts are equally likely | Two fair flips give counts with weights 1/4,1/2,1/4 | List atomic outcomes before grouping |
| Sensitivity equals precision | Worked detector: .9 sensitivity but precision 90/585 | Label both denominators in a count table |
| Disjoint positive-probability events are independent | Their intersection is zero, their marginal product positive | Test the product definition |
| Duplicate evidence gives another likelihood multiplier | Repeating the same observation supplies no new event | Write the likelihood conditional on earlier evidence |
| A retry cap means a fixed number of trials | First-attempt success stops the experiment | Enumerate terminal paths and count reached attempts |
| Sampling without replacement is binomial | Six machines, two faults, two draws: variance16/45, not4/9 | Compare PMFs, not just equal means |
| Linearity needs independence | Empty-server indicators are dependent but their expectations still add | Sum indicator expectations with finite means |
| Zero covariance proves independence | Symmetric X on[-1,1], Y=X²: covariance zero but deterministic dependence | Distinguish linear association from independence |
| Density height is point probability | Uniform[0,.1] has density10, point probabilities zero | Integrate over an interval and check units |
| Weighted p99 values give overall p99 | 99% at10 ms,1% at500 ms: p99=10 ms,mean14.9 ms | Mix CDFs before taking an inverse |
| The CLT guarantees an accurate result at n=30 | Highly skewed finite-variance populations can converge slowly | State assumptions and distinguish limiting from finite-n claims |
| Duplicating observations reduces uncertainty | The copied-row average is exactly the original average | Keep covariance terms or use the true sampling unit |
| Zero observed failures proves zero risk | At p=.1, twenty nonfailures occur with probability .9^20 | Use a justified interval and disclose assumptions |
| Nominal95% means exact finite-sample95% coverage | Wald at n=1,p=.5 has coverage zero | Enumerate counts and compare actual coverage |
| Faster aggregate mean proves a faster version | Lab D reverses direction under equal workloads | Standardize target weights and plan a causal comparison |
Practice Mistake Checks
After practice A, explain why observing a randomly selected head differs from learning that at least one head occurred. After practice B, explain why a retry cap does not determine the number of executed trials. After practice C, show why copied measurements cannot create independent evidence. Use the counterexamples above to check the explanation, then solve a changed case without the table.
For each error record the original claim, the violated assumption, a corrected derivation, and a changed case solved without assistance. Link it to PS1–PS8 in the assessment map. Use the separate solutions after an attempt, and record any help received.
Before advancing, repair critical errors in conditioning, stopping, dependence, coverage, and causal interpretation. Passing an arithmetic check does not waive these requirements.