Chapters 1 and 2 were about what a sample licenses you to believe. This one is about what to do. Expected value turns a spread of possible outcomes into a single number you can act on. It is the right tool more often than intuition allows, and it fails in a specific, recognisable way. Its most useful consequence is that the best policy usually accepts some error on purpose.
Evans and Rosenthal open their chapter with two readings of the same number (p. 129). The first is an average: if a quantity is 0 half the time and 10 the other half, its expected value is 5; if it is 6 a third of the time and 15 otherwise, it is 12 COMPUTED. The second is a price: a ticket paying a random amount is worth, to a fair gambler, its expected value and no more. The definition that makes both precise (p. 130) is the probability-weighted sum
The price reading is the one that makes expectation a decision rule. It says a choice can be judged before its outcome is known, by averaging over what might happen. It is also the reading the authors immediately warn has “certain limitations”, and point to Example 3.1.12.
Flip a fair coin until it lands heads. If Z tails came first, you win 2Z pennies (Example 3.1.12, p. 133). Each possible outcome contributes exactly half a penny to the expected value — 2z · (1/2)z+1 = 1/2 — and there are infinitely many outcomes. The sum does not converge. By the fair-price reading you should pay any amount at all to play.
Nobody would, and the next example (3.1.13) shows why the refusal is reasonable. Cap the prize at 230 cents, just over $10.7 million, and recompute:
| game | expected value |
|---|---|
| uncapped | infinite |
| capped at 230 cents ($10,737,418.24) | 16 cents |
Almost the entire infinite value lives in outcomes so rare, and so large, that no one could ever collect them COMPUTED. The lesson is not that expectation is wrong. It is that an expected value dominated by its tail is a statement about the tail, and the first question to ask of any expected value is whether the outcomes carrying it can actually happen.
Most decisions made from data are threshold decisions: flag the case or pass it; call the result significant or not; accept the claim or send it back. Each has two ways to be wrong. A false alarm flags something that was fine. A miss passes something that was not. Moving the threshold trades one error for the other, and no threshold removes both while the two populations overlap.
Expected value settles where to put it. Give each error a cost, weight by how often each occurs, and choose the threshold that makes the expected cost smallest. Take a simple model MODEL: real cases score two standard deviations above ordinary ones, one case in ten is real, and a miss costs five times a false alarm.
| policy | false-alarm rate | miss rate | expected cost |
|---|---|---|---|
| flag everything — never miss | 0.999 | 0.000 | 0.899 |
| flag nothing — never false-alarm | 0.000 | 1.000 | 0.500 |
| best threshold, t* = 1.29 | 0.098 | 0.240 | 0.208 |
The best policy misses about one real case in four and raises a false alarm about one time in ten, and it costs less than half of either “perfect” policy COMPUTED. Driving either error to zero is not rigour; it is paying the whole cost of the other error. The same algebra gives the optimum in closed form, t* = d/2 + ln[(1−p)·cFA / (p·cmiss)] / d, and the script checks the formula against a direct search.
What moves the optimum is the ratio of costs. As a miss becomes dearer the best threshold drops and false alarms rise, but at no finite ratio does either error vanish:
| a miss costs | best threshold | false alarms | misses |
|---|---|---|---|
| 1× | +2.10 | 0.018 | 0.539 |
| 5× | +1.29 | 0.098 | 0.240 |
| 20× | +0.60 | 0.274 | 0.081 |
| 100× | −0.20 | 0.581 | 0.014 |
This series runs its own threshold decisions all the time: scanners that flag a sentence as a claim, gates that flag a Lean file as unsafe, correction-box finders, duplicate detectors. The handoff of 2026-09-20 records five of them in one day that “mistook a mention for a claim” — five false alarms. The reflex after a false alarm is to tighten the instrument until it stops. This chapter says what that reflex costs.
A scanner tightened until it never cries wolf has moved its threshold to the right of the table above, and it now misses real claims silently. Misses are the expensive error here: a false alarm costs a minute of reading, while a missed claim gets published. On the table's logic that points to a low threshold and a tolerated stream of false alarms. The right response to a false alarm is to make it cheap to dismiss, not to make it impossible. MODEL — the corpus's scanners have not had their error rates measured, and the numbers above are the model's, not theirs. OPEN Measuring them is the natural next instrument: run each scanner over a hand-labelled sample and report both rates.