M01 · optional contextual support
Count rates and price mistakes
Move between counts, fractions, percentages and expected error costs.
Start with arithmetic, percentages and basic algebra. Open any section directly. Practice and completion here do not change any AI course requirement or the original five-lesson mathematics pilot.
1. Try a concrete case
For 100 items with 20 relevant cases, 90% sensitivity misses 2. A 10% false-alarm rate on the other 80 creates 8 false alarms. If a miss costs 5 and an alarm costs 1, the expected cost is 18.
2. Predict before changing a value
Change false-alarm rate alone, then population size. Explain why doubling population doubles expected totals but leaves per-item cost unchanged. Name the quantity you expect to change and one you expect to stay fixed before using the controls.
3. Experiment
Change one quantity, follow every calculation
- Expected false alarms
- 8.00000
- Expected misses
- 2.00000
- Weighted cost
- 18.00000
Vertical axis: Expected cost per item. Fixed scales; 0 points clipped for display. Lines join sampled values to aid reading; they do not create extra observations. Exact numerical values are below.
| Group | Expected count | Cost each | Subtotal |
|---|---|---|---|
| False alarm | 8 | 1 | 8 |
| Miss | 2 | 5 | 10 |
Assumed prevalence 20%, sensitivity 90%; expected counts may be fractional. Error costs omit human and operating overhead.
Change false-alarm rate alone, then population size. Explain why doubling population doubles expected totals but leaves per-item cost unchanged.
Controls save locally as practice. No timers, remote services or GPU are required. Maximum numerical work is a bounded small table or 256 bootstrap resamples.
4. Understand the calculation
Start with a calculation you can inspect
For 100 items with 20 relevant cases, 90% sensitivity misses 2. A 10% false-alarm rate on the other 80 creates 8 false alarms. If a miss costs 5 and an alarm costs 1, the expected cost is 18.
Move between counts, fractions, percentages and expected error costs. The experiment changes one named quantity at a time and shows the intermediate values, rather than asking you to accept a final score. Before changing a control, write down which value you expect to move and why.
Read the formula in words and symbols
expected cost = false alarms × alarm cost + misses × miss cost
A rate is a count divided by the relevant group size. 10% = 10/100 = 0.1. Costs use fictional credits; prevalence and sensitivity are assumptions.
A subscript names a coordinate, case or time index; it is not multiplication. A summation sign means add the terms for the stated indices. Parentheses identify which calculation happens first. Carry units through arithmetic: adding incompatible quantities or confusing a rate with a count changes the question.
Connect the visual and the numbers
Change false-alarm rate alone, then population size. Explain why doubling population doubles expected totals but leaves per-item cost unchanged. The plot uses declared fixed axes, so a new setting does not silently rescale the apparent change. Any clipped values are counted and remain available in the numerical table. Do not infer an exact curve or a new observation from a line drawn between points.
Use it in an AI experiment
These ideas support C01, C02, C08, C09. Use the return link when you entered from an AI lesson; its original section and experiment remain separate. A mathematical support calculation is a way to inspect an assumption, not evidence that a trained model generalizes.
Optional derivation and implementation depth
Write the weighted sum as an expectation over mutually exclusive outcomes. Add human review and operating costs only when their assumptions are explicit. A loss function may choose different weights; it is not automatically a measured financial loss.
Recompute the worked example with a separate piece of arithmetic or a short program. State the domain and test a boundary case before using the formula elsewhere. Guided numerical practice, independent implementation and human review are different outcomes.
Optional implementation and research depth
Write the weighted sum as an expectation over mutually exclusive outcomes. Add human review and operating costs only when their assumptions are explicit. A loss function may choose different weights; it is not automatically a measured financial loss.
Recompute one row independently, then test a boundary case and state which assumption it probes. Passing the supplied practice case does not establish independent implementation proficiency.
5. Apply it to a different case
Practice not yet passed; independent mastery and human review are not claimed.
6. Continue when useful
Completion not recorded. Skipping retains settings and practice; it never awards completion.
Used in C01, C02, C08, C09. Browse every support topic and course connection.