A campus team asks whether an extra shuttle should run during one busy hour. “Improve transport” is too broad to determine a mathematical model. Does improvement mean shorter average waits, fewer missed connections, lower cost or better access for the least-served stop? Write one decision and one measurable output before choosing an equation.
All numerical scenarios in this unit are original synthetic teaching cases. They illustrate reasoning, not calibrated claims about a real campus, business or population. The unit prepares a substantial investigation; it does not itself provide independent capstone review.
Predict
Suppose an extra shuttle costs 8 units and prevents a disruption costing 40 units whenever that disruption would occur. Predict the probability threshold at which the intervention becomes worthwhile under expected monetary loss. Then explain why a different stakeholder might reasonably choose another loss measure.
Experiment
Let p be the disruption probability. Doing nothing has expected loss 40p; the intervention has loss 8 under the stated perfect-prevention assumption.
p
No-intervention expected loss
Intervention loss
Preferred under this criterion
1/10
4
8
No intervention
1/5
8
8
Tie
2/5
16
8
Intervention
Change perfect prevention to prevention with probability 3/4 conditional on disruption. The intervention now has expected loss 8+10p. The threshold becomes 8/30=4/15. Identify exactly which assumption changed and which arithmetic changed with it.
Understand
A question contract records the decision alternatives, target population, time horizon, output, units, data and acceptance criterion. It also records exclusions. For example, a one-hour steady-capacity model may exclude overnight staffing and unusual events. Those exclusions limit use; they are not evidence that unusual events never matter.
Prediction and decision are different layers. A model estimates p or a distribution of waiting times. A decision rule combines that information with losses, constraints and preferences. The same prediction can support different actions under different loss tables. Conversely, a decision can be insensitive to moderate prediction error if every plausible p lies far from the threshold.
Make validation falsifiable: specify an error statistic and tolerance before seeing the evaluation observations. “The graph looks reasonable” leaves no clear failure condition. A useful statement might restrict the target hour and stops, reserve future observations and compare mean absolute error with a chosen tolerance, while reporting uncertainty from the small sample.
Official breadth references include Boston College MATH4470, whose undated page emphasizes model construction and discrete systems, and Duke MATH477S, Spring 2021, which connects individual models with mathematical writing. They inform this original sequence, not an equivalence claim.
Apply
Write a contract for one school, college or workplace question. Include two feasible alternatives, a measurable output and a condition under which the model should not be used. Calculate one decision threshold from an explicit loss table. State which quantities are assumptions and which would need observations before a real recommendation.
Continue
Next, derive a balance equation from a system boundary. Keep the question contract nearby: a mathematically correct equation can still answer the wrong question if its output does not match the intended decision.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A plan has failure probability 1/5, failure loss 30 and fixed cost 2. What is its expected total loss?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
An intervention costs 6 and completely avoids a loss of 24 when failure occurs. At what failure probability is it tied with doing nothing?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Turn “improve campus transport” into a bounded mathematical question.
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.
Explain why the most accurate probability forecast need not imply the cheapest action.
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.
Write a validation claim that could fail for a delivery-time model.
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.
Draft an investigation contract for one school, college or workplace decision.
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.
