Did you know that a warning system can correctly recognize nine out of ten genuine problems and still be wrong more often than right when it raises an alarm? The missing piece is the base rate: how common the problem was before the warning arrived. Intelligent judgment combines the event’s prior frequency with the warning’s sensitivity and false-alarm rate. It does not turn a strong-sounding test score into a probability that a particular alert is true.
This guide teaches the idea with a fictional operations audit, not real student, medical or fraud findings. You will build a complete count table, calculate what a positive signal actually means, see why a rare problem creates many false positives, and learn how to update responsibly when the population or warning system changes. The goal is practical reasoning for learners, teams and organisations: better questions about denominators, evidence and decisions before anyone acts on a red flag.
Contents
- What a base rate tells us
- The four cells of an outcome table
- A complete fictional 10,000-case example
- Why the false alarms outnumber true alarms
- Bayes theorem from counts rather than mystery
- Odds and likelihood ratios offer another view
- The same warning can have a different meaning elsewhere
- Rarer events make precision collapse
- Reference classes are choices that require evidence
- An alarming story is not a denominator
- Sensitivity and precision create trade-offs
- False positives carry human costs
- False negatives also carry costs
- When accuracy conceals the problem
- The prior can change over time
- Sequential evidence is not automatically independent
- A transfer example for an unfamiliar population
- Communicating uncertainty to other people
- What organisations should measure
- Diagnosing a bad inference
- Limits of base-rate reasoning
- Frequently asked questions
- Reader pathways and primary references
What a base rate tells us
A base rate is the frequency of an event in a relevant population before considering a new observation. If two out of every hundred inspected transactions contain a specified issue, the prevalence or prior probability in that setting is 2%. That does not imply that every particular transaction has a 2% chance in all circumstances, but it gives a population-level starting point. The new signal must be interpreted against that starting point to support a revised estimate.
People often pay more attention to vivid case descriptions, alarming messages or impressive test performance than to quiet background frequencies. A statement such as ‘90% of real defects are caught’ concerns a different conditional probability from ‘90% of alarms indicate real defects’. Confusing those statements is a form of base-rate neglect. The first asks how an indicator behaves among true cases; the second asks how true cases are represented among all indicators.
Reliable reasoning begins by identifying the reference class. A screening procedure used on a selected high-risk population may have a very different positive predictive value from the same procedure used on a broad population with fewer problems. The appropriate base rate is specific to the population, time period, criteria and outcome being analyzed. If these are unknown, a point estimate should be presented as conditional on assumptions rather than as certain.
The four cells of an outcome table
Most binary warning systems can be described by four counts. A true positive is a real event correctly flagged. A false negative is a real event not flagged. A false positive is a non-event flagged as if it were an event. A true negative is a non-event correctly left unflagged. These counts refer to a declared outcome definition and reference standard; changing that definition changes the table even if the same warning program runs.
Sensitivity is true positives divided by all actual events: TP/(TP+FN). Specificity is true negatives divided by all actual non-events: TN/(TN+FP). Positive predictive value, sometimes called precision in machine learning, is true positives divided by all positive warnings: TP/(TP+FP). Negative predictive value is true negatives divided by all negative warnings: TN/(TN+FN). These denominators are not interchangeable.
To ask ‘how often does the warning catch a real problem?’, use sensitivity. To ask ‘how often is a warning justified?’, use positive predictive value. To ask ‘how often are clear cases really clear?’, use negative predictive value. To ask ‘how frequently does the warning fire on normal cases?’, use the false-positive rate FP/(FP+TN), which equals one minus specificity. Each question refers to a different slice of the same table.
A complete fictional 10,000-case example
Imagine a made-up internal audit of 10,000 digital submissions. For this teaching example alone, suppose 2% contain an independently verified formatting defect. There are therefore 10,000 × 0.02 = 200 truly defective submissions and 10,000 − 200 = 9,800 non-defective submissions. Assume an automated flag is 90% sensitive and 95% specific for this exact defined defect. These are invented parameters, not performance measurements of any existing tool.
Among 200 genuinely defective submissions, sensitivity of 90% produces 200 × 0.90 = 180 true positives and 200 − 180 = 20 false negatives. Among 9,800 normal submissions, specificity of 95% produces 9,800 × 0.95 = 9,310 true negatives and 9,800 − 9,310 = 490 false positives. The counts reconcile: 180 + 20 + 9,310 + 490 = 10,000. There are 180 + 490 = 670 positive warnings and 20 + 9,310 = 9,330 negatives.
The chance that a flagged case genuinely has the defined defect, using these assumed population parameters, is 180/670 ≈ 0.2686567, or 26.87%. The chance that a negative case is actually normal is 9,310/9,330 ≈ 0.9978564, or 99.79%. Notice the contrast: a system can have high sensitivity and specificity while most of its positive flags are false in this particular low-prevalence population. That does not make the system worthless, but it changes how warnings should be handled.
Why the false alarms outnumber true alarms
In the fictional example, only 200 of 10,000 cases truly have a defect while 9,800 do not. Even a relatively small 5% false-positive rate applied to the much larger normal group creates 490 false alarms. Meanwhile, catching 90% of the much smaller defective group creates only 180 true alarms. The difference comes from multiplying two conditional error rates by groups of very different sizes.
An easy mental shortcut is to build a concrete hundred, thousand or ten-thousand-case table rather than memorizing Bayes’ theorem as symbols. First count how many cases really have the outcome. Apply sensitivity to that group and false-positive rate to the other. Only then divide true alarms by all alarms. A calculator is useful for decimals, but the hardest reasoning step is often choosing the right denominator and population.
Do not say ‘the system is only 26.87% accurate’. Accuracy would be (TP+TN)/N = (180+9,310)/10,000 = 94.9% in this made-up case, and that aggregate can also be misleading. Positive predictive value, sensitivity, specificity and accuracy answer different questions. Use the metric whose denominator matches the decision, and present the other critical rates to avoid hiding costs.
Bayes theorem from counts rather than mystery
Let P(D) mean the prior probability of the defined event, P(+|D) the chance of a positive warning when the event exists, and P(+|not D) the false-positive rate. Bayes’ theorem yields P(D|+) = P(+|D)P(D) / [P(+|D)P(D) + P(+|not D)P(not D)]. This is the same count argument expressed in probability notation. The denominator is the total probability of a positive warning from both event and non-event groups.
Substitute the fictional numbers: P(D)=0.02, sensitivity=0.90, false-positive rate=0.05, and P(not D)=0.98. The numerator is 0.90 × 0.02 = 0.018. The denominator is 0.018 + 0.05 × 0.98 = 0.018 + 0.049 = 0.067. The posterior probability is 0.018/0.067 = 18/67 ≈ 0.2686567. Multiplying these probabilities by 10,000 reproduces the 180 true warnings and 670 total warnings.
This identity lets us check a numerical answer in two independent ways. If the count-table result and probability formula disagree, inspect the prevalence, the denominator and whether the false-positive rate was confused with specificity. A good explanation makes the units visible: the numerator is the proportion of all cases that both have the issue and are flagged, while the denominator is the proportion of all cases that are flagged.
Odds and likelihood ratios offer another view
Instead of directly computing the posterior probability, Bayes’ rule can update odds. Prior odds are P(D)/(1−P(D)); for a 2% base rate, they are 0.02/0.98 = 1/49 ≈ 0.020408. A positive likelihood ratio is sensitivity divided by false-positive rate. Here it is 0.90/0.05 = 18. Posterior odds are (1/49) × 18 = 18/49 ≈ 0.367347. Converting odds to probability gives (18/49)/(1+18/49) = 18/67 = 26.87%.
This method separates how common a condition is from how discriminating the evidence is. A likelihood ratio of 18 says a positive warning is 18 times as likely under the event condition as under the no-event condition, according to assumed rates. It does not say there is an 18-to-1 chance that the warning is correct. Before multiplying, the prior odds already strongly favor ordinary cases because the event is rare.
Negative results also update odds. The negative likelihood ratio is (1−sensitivity)/specificity = 0.10/0.95 ≈ 0.105263. Starting from prior odds 1/49 gives posterior odds of an event after a negative warning of roughly 0.002148. That corresponds to an event probability of 20/(20+9,310) ≈ 0.2144%, which is one minus the negative predictive value. The full table and odds method agree.
The same warning can have a different meaning elsewhere
Now move the invented warning system to another population where 10% of cases have the same defined defect, while retaining 90% sensitivity and 95% specificity as a teaching assumption. Among 10,000 cases there are 1,000 defects, giving 900 true positives and 100 false negatives. Of 9,000 normal cases, 5% produce false warnings, giving 450 false positives and 8,550 true negatives. There are 1,350 positive warnings.
The new positive predictive value is 900/(900+450) = 900/1,350 = 66.67%. Nothing about the warning’s sensitivity or specificity changed; only the prior event frequency changed. This is a useful antidote to the statement ‘a positive test always means the same thing’. The probability that matters after a warning depends on how likely the condition was in the relevant group before the warning.
Real systems may also change sensitivity and specificity across populations because case severity, instruments, data quality or user behavior differ. The fixed rates above are assumptions for isolating one effect, not a promise about deployment. A transfer from one population to another requires evidence that the warning mechanism remains valid, plus a trustworthy new base-rate estimate.
Rarer events make precision collapse
Consider a third fictional population of 100,000 cases in which only 0.5% have the defect. That is 500 truly defective and 99,500 ordinary. Under the same assumed 90% sensitivity and 95% specificity, we get 450 true positives, 50 false negatives, 99,500 × 0.05 = 4,975 false positives and 94,525 true negatives. Total positive warnings are 450 + 4,975 = 5,425.
The probability that a positive warning corresponds to a real defect becomes 450/5,425 ≈ 8.29%. The count reconciliation still works: 450 + 50 + 4,975 + 94,525 = 100,000. A system that seemed to produce one correct warning for roughly every three positives in the first population now produces fewer than one correct warning for every ten positive flags. Again, there is no paradox. A large base of normal cases creates a substantial number of false positives.
This matters when organisations expand screening from a targeted group to an entire population. A threshold chosen for a narrow audit can inundate a broad review queue with false warnings. Before rollout, forecast absolute alert volume and the burden on people who must inspect it. A percentage without a denominator does not reveal how many real individuals will be affected.
Reference classes are choices that require evidence
Which base rate is correct? It depends on the class to which the case belongs and how that class was assembled. A school-wide rate may differ from a rate for a particular task type, and a weekday operating process may differ from a weekend emergency procedure. Narrowing the class can improve relevance if there is enough sound data and the distinction is established before inspecting the outcome. Narrowing it after seeing a surprising answer may simply produce a convenient story.
An appropriate reference class balances comparability with statistical stability. A sample of twelve cases might be closely matched but too small for a dependable rate. A million-case historical dataset might be precise about a population that no longer exists. Look at the collection process, timestamps, inclusion criteria, labels and uncertainty. If the base rate is estimated rather than known, use a plausible range and show how conclusions vary across that range.
Beware of selection effects. If only cases that received an earlier warning were checked by experts, the observed proportion of defects among reviewed cases is not the prevalence among all cases. Missing verification for unflagged cases can bias both the base rate and sensitivity estimates. A careful analyst distinguishes population prevalence from the fraction of problems in a selected audit sample.
An alarming story is not a denominator
Human attention naturally favors a vivid example. One spectacular false alarm, widely shared online, may create a feeling that all alarms are useless. One striking genuine discovery may make an indicator look nearly infallible. Neither story tells us how many total alarms occurred or how many events went unnoticed. Intelligent evaluation asks for the full outcome table rather than choosing a conclusion from the most memorable anecdote.
Teams can reduce this mistake by demanding four counts whenever a binary classification is used in a material decision. If some counts are unknown, write unknown rather than guessing. Ask: how many events existed, how many were caught, how many ordinary cases were falsely flagged, and how were the unflagged cases checked? This question often uncovers more than a presentation slide showing a single attractive accuracy percentage.
The broader How Intelligence Works guide connects attention, evidence and judgment. Base-rate reasoning is a particular numerical discipline within that wider map: it forces the invisible background population back into view when a specific signal captures attention.
Sensitivity and precision create trade-offs
A warning threshold can often be adjusted. Lowering it may catch more real issues but also flag more normal cases. Raising it may reduce review burden at the cost of more missed problems. Such changes alter sensitivity and false-positive rate together, not in a fixed universal way. Which trade-off is appropriate depends on the cost of missing a real issue, the harm of flagging a normal case and the availability of safe independent review.
Return to the fictional 10,000-case population with 200 real defects. Suppose a stricter threshold changes sensitivity from 90% to 85% and specificity from 95% to 98%. Then true positives are 170, false negatives 30, false positives are 9,800 × 0.02 = 196, and true negatives are 9,604. The flagged total becomes 170 + 196 = 366. Precision improves to 170/366 ≈ 46.45%, while 10 more genuine cases are missed than before.
The stricter threshold generates 304 fewer warnings than the original 670, but catches 10 fewer actual defects. This is not automatically better or worse. A low-risk classroom feedback tool might prefer minimal disruption, while a critical safety system may require strong sensitivity and qualified follow-up. Do not choose an operating threshold by precision alone. Compare absolute consequences and verify assumptions with held-out or future data.
False positives carry human costs
In a review system, every false positive consumes attention and can unfairly affect a person whose work is legitimate. Even if the workflow requires a human before action, a flood of weak warnings can overload reviewers, encourage rubber-stamping or damage trust. Present flagged cases as candidates for examination, not established wrongdoing. A posterior probability is a model of uncertainty, not a moral verdict.
An invented cost example illustrates the load. If an organisation pays $5 in staff time to review each of the original 670 warnings, the direct review cost is 670 × $5 = $3,350. If its stricter threshold produces 366 warnings, review time at the same unit cost would be 366 × $5 = $1,830, a difference of $1,520. These figures omit harm from missed defects and any benefit from finding a genuine one; they cannot establish the better policy alone.
Reliable process design separates signal generation, verification, escalation and decision. Maintain an appeal and correction route where people’s opportunities or reputations could be affected. Track false alarms by group and context. A classifier that appears effective overall may generate a disproportionate review burden for one subgroup, so aggregate positive predictive value is not the only fairness-relevant measure.
False negatives also carry costs
A false negative is a real event that a warning system fails to detect. A high negative predictive value in a low-prevalence population does not mean false negatives are harmless. It means that most unflagged cases are ordinary, which was already likely because the event was rare. Whether the missed cases matter depends on the stakes and any additional safeguards.
Return to the 10,000-case example. There are 20 false negatives among 9,330 negative outcomes. The residual event rate among negative outcomes is 20/9,330 ≈ 0.2144%. This sounds small, but a process with severe consequences may still require targeted checks, monitoring or alternative confirmation. A blanket statement that ‘negative means safe’ would overstate the evidence.
Intelligent systems should distinguish probability from consequence. If one event can cause a disproportionate loss, a decision may warrant caution even at a low probability. Equally, costly interventions justified only by weak warning evidence can create damage of their own. The appropriate action results from combining calibrated uncertainty with values, costs, safeguards and domain expertise.
When accuracy conceals the problem
Suppose a trivial classifier in the 2%-prevalence fictional population predicts ‘no defect’ for all 10,000 cases. It is right on 9,800 normal cases, so its accuracy is 9,800/10,000 = 98%. Yet it catches none of the 200 defects; its sensitivity is 0%, and it produces no positive warnings to evaluate. Calling it a 98%-accurate detection system would obscure its complete failure at the task people cared about.
By comparison, the hypothetical 90%-sensitive and 95%-specific flagger had 94.9% accuracy while detecting 180 true defects. Which is preferable depends on how missed defects and unnecessary reviews are valued. The comparison shows why the evaluation must be anchored to the reader’s actual purpose rather than one headline metric. In rare-event detection, sensitivity, precision, false-positive rate, absolute workload and cost often matter more than raw accuracy.
The same reasoning applies to forecasts of outages, unusual transactions, quality failures or other uncommon events. Before rewarding an algorithm for a high percentage correct, calculate the prevalence and compare it with a simple baseline that always predicts the common class. A sophisticated model should demonstrate useful improvement in the decisions it enables, not just on a benchmark’s easiest count.
The prior can change over time
Base rates are not permanent features of the universe. An operational control can improve and reduce genuine defects, a new process can introduce unfamiliar failure modes, or a seasonal surge can increase risk. If a warning system holds the same sensitivity and specificity while prevalence changes, its positive predictive value changes as the earlier examples show. A report based on last year’s population should not be projected without checking whether the relevant rate has shifted.
Updating a prior requires trustworthy recent evidence. A sudden increase in detected warnings may be caused by more real events, a threshold change, an instrument error, a new data source or ordinary variation. Those causes lead to different decisions. Measure underlying outcomes with an independent reference wherever possible instead of treating the number of warnings as a direct measure of prevalence.
Do not silently replace a previously documented prior to produce a more satisfying conclusion. Version the population definition, evidence period, sample size and uncertainty range. If several plausible priors remain, report a range of posterior probabilities and what decisions would change across that range. That is more honest than a single decimal place that pretends away missing information.
Sequential evidence is not automatically independent
Suppose two tools both raise the same warning. It can be tempting to multiply their likelihood ratios as if two votes necessarily provide twice the evidence. That only works under relevant conditional-independence assumptions or a correctly modeled joint distribution. Two models trained on the same labels and using the same input feature may make highly correlated errors. A second alert can largely repeat the first rather than add independent information.
An especially risky practice is to run one piece of evidence through several paraphrasing systems, count each restatement as an independent confirmation, and conclude that the probability has skyrocketed. Corroboration means distinct evidence with known relationships, not multiple descriptions of a shared observation. A checklist should ask what each test actually measured, which causes of error they share and whether they were calibrated for sequential use.
The National Academies medical diagnostic technology assessment available through NCBI Bookshelf discusses assumptions behind combining test results and warns that conditional independence may fail. Its example domain is medical testing; the general lesson travels to organizational warnings. Do not import clinical performance rates into unrelated datasets.
A transfer example for an unfamiliar population
Now a hypothetical university administrator receives the same 90%-sensitive, 95%-specific warning system but learns that the relevant audited issue occurs in only one out of 200 cases. In a teaching population of 20,000, one out of 200 means 100 actual issue cases and 19,900 normal cases. Sensitivity yields 90 true positives and 10 false negatives. A 5% false-positive rate yields 995 false positives and 18,905 true negatives.
There are 90 + 995 = 1,085 positive warnings. The chance a flagged case actually has the issue under these assumed conditions is 90/1,085 ≈ 8.29%. The correct denominator is all positive warnings, not 20,000 total cases, not 100 true cases and not 1,085 plus all negatives. The counts reconcile: 90 + 10 + 995 + 18,905 = 20,000. The residual issue rate among negatives is 10/(10 + 18,905) ≈ 0.0529%.
The transfer lesson is to rebuild the four-cell table whenever the reference population changes. Memorizing ‘90% sensitivity and 95% specificity’ is not enough. The administrator should also ask whether those assumed rates were measured in a comparable population and what verification process will protect anyone whose work is flagged. The numerical exercise demonstrates reasoning, not a license to make unreviewed high-stakes judgments.
Communicating uncertainty to other people
A report that says ‘almost 27% of flags correspond to true defects under the assumed rates’ is more useful than ‘the tool is 95% reliable’. Specify the baseline population and the period from which it was estimated. Distinguish assumed parameters from independently measured performance. When sample sizes are limited, show uncertainty intervals or plausible ranges rather than treating estimated rates as exact constants.
Natural-frequency statements often communicate the logic better than abstract probabilities. In the worked example: out of 10,000 cases, 200 really have the defect; 180 of those are flagged; 490 ordinary cases are also flagged; therefore 180 of 670 warnings correspond to true defects. A reader can check every link in that chain. This also gives administrators an immediate idea of review burden.
Use respectful language. A ‘flagged’ person is not a ‘guilty’ person; a system may flag a record, not determine intent. For scientific or medical testing, probabilities require qualified interpretation and cannot substitute for professional evaluation. This article’s cases are deliberately nonclinical fictional audits and should not be treated as individualized risk advice.
What organisations should measure
Before deploying an indicator, define the outcome against an independent standard and collect a representative evaluation set. Measure not only true and false positives but false negatives, total tested, prevalence and unknown outcomes. Document how ambiguous cases were handled, who reviewed them and whether review itself can introduce bias. A model evaluated only on previously flagged cases may seem precise while hiding its missed detections.
After deployment, monitor how often the indicator fires, whether the underlying population changes, how long reviews take and how often reviewers overturn warnings. Segment rates thoughtfully where material risks justify it, but avoid revealing personal information or making claims based on tiny unstable groups. If an alert volume rises sharply, investigate changed prevalence, instrumentation and selection effects before changing thresholds.
A periodic audit should include some unflagged cases so that false negatives are not invisible. Independent review costs resources, so choose a sampling plan appropriate to the decision’s stakes and report its coverage. A human approval process without adequate evidence and time is not automatically a reliable safety mechanism. The system needs a mechanism to learn from both confirmed warnings and detected misses.
Diagnosing a bad inference
If someone concludes ‘90% of alarms are correct because sensitivity is 90%’, ask which denominator they used. Sensitivity divides confirmed caught events by all actual events. Positive predictive value divides caught events by all warnings. Show the full count table to repair the error. If a report claims 95% specificity means a 5% chance a positive is wrong, explain that 5% is the false-positive rate among normal cases, not among positive warnings.
If someone gets a different posterior after changing the population, do not automatically call the result inconsistent. Recalculate with the new prevalence. If the numerator and denominator appear impossible, perform count reconciliation: the four cells should add to N, each conditional denominator should refer to the correct group, and no category should be counted twice. Fractions with small imaginary populations can also create fractional persons; use a sufficiently large cohort when teaching exact whole-number counts.
If two warnings are treated as independent without evidence, ask whether they share inputs, training labels, analysts or instruments. If actual outcomes were verified only after positive warnings, ask how sensitivity and prevalence were estimated. If a system’s final recommendation depends on the costs of different mistakes, ask whose values and obligations were included. These questions separate a computation error from an evidence-quality problem or a decision-policy problem.
Limits of base-rate reasoning
Knowing the base rate does not reveal an individual’s complete situation, and a posterior probability is not an oracle. The formulas depend on correctly defined events, representative populations, trustworthy sensitivity and false-positive estimates, and appropriate assumptions about evidence. A rate measured on an old or selected sample can lead to confident-looking but inaccurate conclusions. Probability calculations cannot correct a biased collection process by themselves.
Binary tables also simplify many real problems. Some outcomes have several severity levels; warning scores can be continuous; experts may disagree about ground truth; and interventions can change future rates. A base-rate calculation is one component of intelligent decision-making, not a substitute for ethical review, causal reasoning or domain-specific safeguards. Where consequences are severe, qualified independent interpretation remains essential.
Finally, frequentist counts and Bayesian updating are tools for uncertainty, not excuses to ignore evidence that disagrees with a preferred prior. If an indicator has highly specific and trustworthy evidence, it can legitimately change beliefs a great deal. Base-rate neglect is repaired by combining relevant prior information with new information, not by always choosing the common explanation.
Frequently asked questions
Why can most flags be false if sensitivity is high? Because normal cases can vastly outnumber real event cases, so even a small false-positive rate generates many warnings. What is the difference between sensitivity and precision? Sensitivity measures the share of real events caught; precision measures the share of warnings that are genuine. Which one matters more? That depends on the costs of missed cases and unnecessary actions; many decisions need both.
Can accuracy alone assess a rare-event detector? Usually not. A rule that predicts the common outcome for every case can look accurate while detecting nothing. Does a higher base rate always increase positive predictive value when sensitivity and specificity stay fixed? For ordinary nondegenerate rates, yes; the worked tables show how the balance of true and false warnings changes. Are two positive tests automatically decisive? No; their errors may be correlated, and calibration or conditional-independence assumptions matter.
Can I use this article to interpret a personal medical result? No. The numerical illustrations are fictional administrative audits, not medical advice. A clinician must account for the actual test, clinical context and individual circumstances. What is the fastest way to avoid denominator mistakes? Write the four counts for a clearly defined hypothetical population and check that they add to the total.
Reader pathways and primary references
If you are learning percentages, reproduce the 10,000-case audit with a table and calculator before using symbolic Bayes notation. If you teach mathematics or critical thinking, ask students to explain why 95% specificity does not mean a positive warning is 95% correct. If you manage a review process, estimate absolute false-alarm workload and build independent verification before changing policy. If you conduct research, state the population, outcome definition and uncertainty of every measured rate.
Primary technical references include the US FDA guidance on reporting diagnostic test results, which defines sensitivity and specificity for its own application domain, and the NCBI Bookshelf assessment of diagnostic testing and Bayes’ theorem, which examines prior probabilities and sequential evidence. We use these for probability principles, not to make claims about an actual diagnosis, test or patient.
Related reading includes information cascades and independent signals, collective forecasting, and the How Intelligence Works hub. For students, the education and learning transfer guide emphasizes moving from a solved classroom example to a changed situation, exactly the skill needed when the base rate changes.