Base Rates, Probability, and Calibrated Judgment

“This screen finds 80% of the companies that later achieve high growth.” Lin is ready to treat its results as promising candidates. A friend asks a different question: Does finding 80% of the high-growth companies mean that 80% of the selected companies will achieve high growth?

Both statements contain the same percentage, but they count different groups. Sorting companies by outcome and then adding the screening flag makes the difference visible. A seemingly strong signal can still produce a long list of companies that do not meet the target.

What you will be able to do

• Explain which groups a base rate and a conditional probability are counting.
• Update a judgment from supplied evidence while recording assumptions that remain unverified.
• Distinguish a single outcome from calibration and an investment decision, and save three forecast records.

How common was the event before the screen?

All companies, counts, rules, and outcomes in this exercise are independent synthetic teaching inputs. They are not historical research results and do not reuse the earlier business cases. Here, high growth means revenue growth of more than 20% in the next complete fiscal year, measured against the preceding year. It says nothing about stock returns.

Suppose we have 1,000 company observations with known outcomes, consistent definitions, and the same observation horizon. Of these, 100 meet the high-growth definition and 900 do not. High-growth observations make up 10% of the group. Before looking at the screening flag, this proportion is our base rate: how common the event is in the specified reference group.

The reference group is part of the evidence. Companies at different stages, with different businesses or economic conditions, may not belong in one comparison. In real research, ask whether failed companies were omitted, whether only survivors remain, and whether the data would have been available at the relevant time.

“It worked eight times out of ten” leaves all of that unresolved. How were those ten cases chosen? Ten handpicked winners cannot establish the base rate for an entire candidate pool. Two independent observations do not make an estimated proportion stable either. A base rate needs a stated scope, source, and limitations; it is a starting point to inspect, not an authoritative number to accept.

Find 80% of the targets, yet fill less than half the list

The screen's results are given. Among the 100 high-growth observations, it flags 80 and misses 20. Among the 900 that do not achieve high growth, it flags 180 and leaves 720 unflagged. You do not need to estimate any missing cell.

“Finds 80% of the high-growth companies” divides 80 by 100. It starts inside the group whose outcomes are already known to be high growth and asks how many the screen identified. Lin, however, receives the flagged list. She has to start inside that group instead.

There are 80 plus 180, or 260 flagged observations. Only 80 achieve high growth. Divide 80 by 260, and the high-growth share is approximately 30.77%. The screen changes the proportion from 10% in the full group to about 30.77% in the flagged group. In these supplied data it provides information, but it does not produce an 80% high-growth list.

The two questions run in opposite directions: “How often is a high-growth company flagged?” and “How often does a flagged company achieve high growth?” Conditional probability measures the event within the group where a specified condition holds. Changing that condition changes which observations belong in the denominator. [1]

Why are there so many unwanted flags? The non-high-growth group starts with 900 observations. Flagging only 20% of that large group still adds 180 names, more than the 80 desired names. A high identification rate cannot, by itself, turn a rare event into the majority of a screening list.

These proportions are exact calculations from the given counts, subject to rounding in their presentation. Applying them to future companies requires more: a suitable reference group, screening performance that transfers, and allowance for estimation uncertainty. Correct arithmetic does not establish those conditions for you.

An update needs evidence for its size

Moving from 10% before seeing the flag to about 30.77% conditional on the flag illustrates an update. Start with how common the event is, examine how often the evidence appears under different outcomes, and revise the judgment. Bayes' theorem connects these conditional probabilities. For this chapter, being able to reconstruct the counts is enough. [1]

Now suppose a news story repeats the very same screening result. It is not a second independent screen. You cannot raise the probability again as though a separate test had confirmed the first. Trace the new item's source before treating it as additional information, just as you traced republications in G12.

An acquisition creates another kind of question. If new evidence shows that reported revenue includes an acquired business, what event were you forecasting: reported total growth, or growth in the original business? The evidence may matter differently to each. You cannot start with the first definition, switch to the second after inconvenient information appears, and claim that your forecast was unchanged.

Real investment questions often lack a complete table of counts. You may be able to state a working range based on comparable company records and identify reasons for moving it up or down. You may instead have insufficient evidence to assign a number. AI should not fill missing detection and false-flag rates with plausible-looking guesses.

A subjective probability expresses a judgment about an uncertain event. A historical frequency is a proportion observed in a particular sample. The two can inform each other. Writing both as percentages does not turn an unmeasured judgment into an observed frequency.

Give each forecast an event and a resolution date

Prepare three blank records without filling in outcomes. The companies and events are fictional:

• Will Songxi's revenue grow more than 20% in its next complete fiscal year?
• Will Qingshi's specified new product begin commercial sales by June 30, 2027?
• Will North Shore report positive operating cash flow in its next complete annual report?

For each record, identify the material available now, your judgment or range, a deadline, and the document that will resolve the outcome. Songxi needs a named fiscal year and revenue scope. Qingshi's commercial sales cannot become an internal product test after the deadline. North Shore's operating cash flow must remain distinct from profit. Define the event before there is a result to interpret.

No supporting inputs have been provided for these three companies. You do not need to manufacture forecasts of 70%, 60%, and 80%. “Comparable revenue is missing,” “commercial sales needs a definition,” and “cash-flow data is missing” are legitimate initial entries when paired with the next material to obtain. Recognizing what you cannot yet estimate is a useful forecast-log result.

Suppose suitable material later arrives and you record a working 60% judgment for Songxi. Consistently defined order evidence then leads you to raise it to 70%. Preserve the 60% entry and its timestamp, and add the new estimate, evidence date, and reason. Replacing the original with 70% after learning the outcome would destroy the comparison you wanted to make.

A forecast about a business event is also separate from a stock purchase. The price might already reflect stronger growth than you expect. Growth might require substantial spending or additional shares. An event probability supplies neither an appropriate purchase price nor the potential loss and household funding constraints needed for an investment decision.

Does one failed 70% forecast prove you were wrong?

Suppose Songxi ultimately misses the growth threshold. A 70% forecast already allowed a 30% chance of that outcome. One failure does not establish that the estimate was invalid. Nor does one occurrence of an event assigned 10% automatically establish that the original estimate was good or bad.

To examine calibration, collect many forecasts made before their outcomes and compare similarly rated events with their observed frequencies. The Met Office's reliability diagrams group forecasts by issued probability and compare them with how often the event occurred. [2] Using that recordkeeping idea in investment research is a teaching adaptation here, not evidence that weather forecasting has validated financial forecasting skill.

If 70 of 100 events assigned 70% occur, that group's observed frequency matches its forecast probability. Three records, even with two occurrences, cannot establish good calibration. Dependence between events, differences in question difficulty, and changing conditions also affect what can be learned from the record.

Calibration is not the whole task. If a class of events normally occurs 10% of the time, repeatedly forecasting 10% might match its long-run frequency without helping distinguish particular candidates. Ask whether the forecasts add information beyond a reasonable base-rate reference and whether that information helps decisions.

Review more than a tally of correct guesses. Preserve failed and unresolved forecasts, then distinguish inadequate evidence, an unsuitable model, an unclear event definition, and an outcome that falls within ordinary uncertainty. After an event, the explanation can feel obvious. A dated log makes the earlier limits visible again.

A range needs a meaning too

You can examine low, middle, and high revenue scenarios for the next year, identifying the sales volumes, prices, or expansion conditions behind each. Three scenarios are not automatically three equally likely futures. The presence of three rows does not justify assigning one-third probability to each.

A revenue range of 9 million to 11 million also needs an explanation and a stated currency or reporting unit. Is this a working scenario range or a probability interval produced through a stated method? Without an appropriate model or reasoning, attaching “95% confidence” does not transform scenarios into a statistical interval. A point estimate, scenario range, and probability interval communicate different claims; the reader needs to know which one you intend.

A major business-model change can make the old reference group less relevant. Adding decimal places to its historical proportion will not repair that mismatch. Revisit the question and record which experience remains useful and which uncertainty cannot currently be quantified. Caution means matching confidence to evidence; it does not mean reporting 50% for everything.

Try the denominators yourself

First explain why 80 divided by 100 is correct but cannot answer the question about high growth within the flagged list. Then identify the information missing from the claim “we detect 80% of the winners” if you want to evaluate the list's composition. Write your answers before reading on.

Try a different candidate pool

A separate synthetic pool contains 1,000 observations, of which 400 meet the same high-growth definition. Assume that the screen still flags 80% of the high-growth group and 20% of the other group. Count the two kinds of flagged observations, then calculate the high-growth proportion in the list. Can you reuse 30.77%? Calculate before reading the answer.

Save three forecasts that can be revisited

Use Songxi, Qingshi, and North Shore, or choose three precisely defined events of your own. For each, record the event definition, judgment date, information cutoff, sources, reference group, subjective estimate or lack of evidence, reasons for upward or downward revision, resolution deadline, and outcome source. Save the initial version now instead of reconstructing it after the result.

Append a row for each later update and preserve earlier values. At resolution, apply the original definition and record occurrence, nonoccurrence, or an unresolved outcome with the remaining evidence gap. Missing data cannot simply be counted as a success or a failure. Ask what sample and grouping would be needed before comparing similarly rated forecasts.

For a self-check, could someone using the same materials understand why you made the judgment? Could new evidence change it while leaving the original visible? A list of conclusions without dates, definitions, and updates cannot support that review.

Keep confidence within the evidence

Fix the event, deadline, and reference group before counting. A screen that frequently identifies targets does not necessarily produce a list dominated by them. The base rate and the number of unwanted flags both shape the answer.

When new information arrives, check its independence and whether it changes the event you defined. Update for a stated reason while retaining the earlier record. One outcome cannot establish calibration, and a probability alone cannot determine a position size. A clear evidence gap gives the next research step something useful to resolve.

References

1. Stanford Encyclopedia of Philosophy: Bayes' Theorem, Summer 2025 archive, section 1. Supports conditional probability and the distinction between opposite conditional questions. The counts and business events here are original synthetic inputs.
2. Met Office: Reliability and Sharpness Diagrams. Supports grouping probabilities against observed frequencies and examining forecast information. Its weather examples are not financial evidence.

Editorial status

AI-assisted draft, synchronized to the accepted second-round Chinese text. Source and arithmetic checks are recorded in the author review. A separate AI reviewer has completed an independent full-text review. Human financial review and target-reader testing remain pending.