祝融说。

Where Should 1.8 Be Multiplied

Tang Ke wrote the historical starting point five-eighths on paper, with the message ratio 1.8 beside it. She tried multiplying directly and got 1.125. Gu Ning pointed at the result exceeding one and said this is not an adoptable probability; both numbers can be traced back to history, yet the connection did not preserve their original meanings.

The 1.8 of the last chapter is the ratio of B's appearance probability under A and under not-A, not an increment factor for the target probability. To connect it to the baseline, one must first make explicit what material the old judgment conditioned on, then, within the same model, let the new observation enter the relative shares of the two outcome classes.

This chapter still works through the fictional Chengwan repair station's eight-request history. We first fully restore these eight items' empirical distribution, then discuss what assumptions applying it to R17 would require. No complete confirmation was obtained for the current client, and the current unknowns are not declared resolved just because the formula can compute.

The difficulty of updating often shows here: the material has provenance, the computation is skilled, and yet a number answering relative support is taken directly as a probability multiplier. A complete computation not only gives the new number but also makes every step state what it targets.

First Explain How the Old Number Was Obtained

A retains the event of receiving complete confirmation of the original first version by Friday 17:00; B retains the condition of receiving the explicit checklist-received message in the earlier window. In the eight-item equal-weight empirical distribution, A's share is five-eighths and not-A's share three-eighths.

The old number in this computation has not conditioned on B. The five confirmations include three with B and two without; all eight enter the starting point. Thus the grouped observation of B can become a subsequently imposed condition, rather than a message already placed in the starting point and then used once more.

No verified initial numerical prediction for R17 existed previously. This chapter uses five-eighths first as the computational starting point of the historical empirical model; treating it as R17's prior probability would require separately positing a transferable assumption—one must not write the paper starting point as a real prediction Gu Ning made at 10:00.

Prior and posterior here state the relative positions before and after the observation enters; the prior is not necessarily a guess with no material, nor is the posterior necessarily closer to the truth. The old judgment may already have used other information, and the new judgment may rely on misreading; reliability still depends on inputs and conditions.

Connecting the Same Observation to the Two Outcome Classes

Within the same model, denote the old probability of A by p, the probability of B given A by q, and the probability of B given not-A by r. The original empirical distribution corresponds to p equal to five-eighths, q equal to three-fifths, and r equal to one-third.

First compute the joint share of A and B: p times q yields three-eighths. Then compute the joint share of not-A and B: one minus p, times r, yields one-eighth. The two do not overlap and together make up the scope in which B is observed.

This step did not add the two likelihoods directly. Each must first be weighted by its corresponding outcome share to enter the same population. Omitting the old shares would covertly replace the actual composition of five and three with a different prior.

The sum of the two paths is one-half, exactly corresponding to the four items with B among the eight. The portion of B's scope belonging to A is three-eighths, so the new probability is three-eighths divided by one-half, yielding three-quarters.

This path table shows what normalization does: first obtain the two parts within the same population, then take the B scope as this round's total scope. It is not an arbitrary final squeeze back inside one to make the numbers look good, but a return to the conditional definition given B.

What the Bayesian Formula Continues Here

When the observation B has positive probability in the model, this update can be written:

$$ P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)} =\frac{qp}{qp+r(1-p)} $$

The first term is the Bayesian formula; the second expands B's probability using A and not-A, which form two mutually non-overlapping classes covering the population. For the formula and the total-probability decomposition, see the Cornell University CS2800 lecture notes on conditional probability.

Its mathematical basis does not require B to precede A in time. But if it is used for ex-ante prediction, B must be obtainable at the time of judgment, and the old probability and the likelihood must also be formed under the same information conditions. The formula permits retrospective description; it does not let retrospective material automatically become ex-ante grounds.

q and r need not sum to one, because they are given within different outcome groups; p and one minus p are the outcome shares complementary within the same old population. If the denominator is zero, it means B has no probabilistic support under this model, and this chapter's ratio update cannot proceed as usual.

Notation can simplify expression but does not replace model choice. If A's event changes, if B's content changes, or if p already includes B, one cannot simply copy the old letters into the formula. A computable form holding still requires checking whether the form corresponds to the material.

How Odds Connect to the Likelihood Ratio

When the probabilities of the target and its negation are both positive, A's odds can be written as P(A) divided by P(not-A). If the probability is p, the odds are p divided by one minus p. This book uses odds for this ratio, still using probability for event shares between zero and one.

The old probability five-eighths corresponds to old odds five to three, that is five-thirds. B's likelihood ratio is nine-fifths; multiplying the old odds by this ratio yields three. New odds of three to one mean A occupies three-quarters of the two classes' total shares and not-A one-quarter.

Written symbolically, when the ratios on both sides are definable:

$$ O(A\mid B)=O(A)\frac{P(B\mid A)}{P(B\mid\neg A)},\qquad P(A\mid B)=\frac{O(A\mid B)}{1+O(A\mid B)} $$

This is another expression, in the same model, of the probability-path computation just made; for the odds form, see MIT course 18.05, Lecture 12. Probability cannot be multiplied directly by 1.8, whereas odds, under the corresponding conditions, can be multiplied by the likelihood ratio.

Both formulations yield three-quarters, which helps check whether arithmetic and scope are consistent; they are not two independent pieces of evidence both validating R17. They depend on the same p, q, and r; if the inputs are flawed, switching to another correct formulation will not eliminate the problem.

Restoring the Original Table Is Not Out-of-Model Verification

Three-quarters happens to equal the frequency of three confirmations among the four B objects in Chapter 7. Because this chapter's p, q, and r all come from the same four-cell table, the formula simply returns, along another path, the correspondence the table already contains—the result's conformity is a mathematical consistency that was due.

One cannot therefore say the Bayesian model has predicted successfully on a new sample. No previously unknown historical outcome was independently tested by this restoration, nor did it bring R17 a confirmation that has occurred. In-sample correspondence and out-of-sample applicability must be stated separately.

This restoration still has practical value. It lets Gu Ning detect the errors of multiplying probabilities directly, omitting the not-A path, or forgetting the prior shares, and see clearly how conditional probability and updating connect. The value belongs to organization and computation, not to a claim of proven empirical performance.

If later an unadjudicated new object's outcome is observed, a separate test can be made. But even one correct guess does not directly prove long-run reliability; Part Four will discuss calibration against prediction groups and outcome records, not replace the whole body of observation here with an answer on paper.

What a Different Starting Point Would Yield

Make an independent hypothetical comparison: keep q at three-fifths and r at one-third, and change only p to one-quarter. The two B paths are three-twentieths and one-quarter; the total B share is two-fifths, and the new A probability is three-eighths.

The message likelihood ratio is still 1.8, tilting relatively toward A, but the new result does not exceed one-half. Relative support and which class ends up with the larger probability are not equivalent; with a different old outcome share, the same evidence may only lift a smaller probability without making it the larger one.

This comparison neither modifies Chengwan's historical p nor supplies another real baseline for R17. It shows that updating cannot be decided by the message ratio alone. If Gu Ning wants to use one-quarter instead, she must explain the new starting point's basis, not pick the p that makes the final number match expectations.

A prior can be revised when new material refutes it, but one must state whether an original input was corrected or a new observation was conditioned on. Writing both as updating easily obscures why the old number changed; preserving the object of revision allows later review of the model instead of only the final probability.

Updating Without B Is Not Subtracting the Same Amount

If the model's observation is not-B, the complements within the same groups are needed. The A path is five-eighths times two-fifths, yielding one-quarter; the not-A path is three-eighths times two-thirds, also yielding one-quarter. After normalization A is one-half.

In odds terms, the old five to three times not-B's likelihood ratio of 0.6 yields one to one, so the probability is one-half as well. The symmetric path shares here are a specific relation of the original table; they do not mean that any message and non-message should move the old probability up or down by the same amount.

In the original table, B moves five-eighths to three-quarters and not-B moves five-eighths to one-half, with the change being exactly one-eighth up and down—but this coincidence must not be generalized into a fixed add-or-subtract rule. Changing the prior or the two classes' message probabilities makes the magnitude of change depend on the new denominator relations.

The R17 main line already has B; this section only demonstrates another observation's update. Objects for which no message material was obtained also cannot be updated as if not-B; not knowing the message status and having adjudicated that the specified window lacked that message must still be kept apart.

Where to Test the Transfer Assumptions

If Gu Ning wants to apply this model to R17, she must at least state that the eight items' historical frequencies represent the current starting point without B, that the two groups' message proportions represent the current correspondence under A and not-A, and that the significant differences not yet handled will not render these inputs useless.

These are application assumptions, not facts of the eight-item count itself. The original reply explicitly says the reception time awaits verification; whether the historical B group all had the same status has not been stated; the four missing registry records and the other four's differing stages and windows have not become clarified because of this chapter's computation.

Gu Ning can display that if these coarse conditional inputs are tentatively adopted, the result is three-quarters, presenting it as a candidate model scenario. She cannot delete the "if adopted" and rename the model scenario as R17's verified final estimate. A mathematically certain result is connected to its conditions; conditional expression is not dispensable politeness.

Domain bridging likewise preserves this boundary. RC's processual completeness supports the representation's ability to be further revised, but does not specify how historical frequencies must transfer; theory reduction explains the uses and limits of compression without manufacturing empirical grounds for q and r.

Check Sensitive Inputs Before Deciding the Adoption Scope

Make an input comparison: keep p at five-eighths and q at three-fifths, and assume only that r equals q, also three-fifths. B is equally easy in both classes, and the update leaves it at five-eighths. It is not that the message failed to arrive, but that this model gives it no relative discriminating role.

If instead r is assumed to be one-fifth, the A-and-B path remains three-eighths while the not-A-and-B path becomes three-fortieths, and after updating A is five-sixths. When r changes, the other path in the denominator changes, and the message's relative support for the target changes too.

These hypothetical computations show that r is an important current input that must not be hidden in the evidential account. But a set of scenarios from five-eighths and three-quarters to five-sixths is not a statistical confidence interval; it has no coverage guarantee, nor proof that R17 must in reality fall within it.

The use of sensitivity comparison is to find which assumption matters to the result, then decide what needs checking or how to narrow the statement's scope. If input changes clearly affect the judgment, they should remain in the delivery; one cannot pick the highest scenario as the conclusion and treat the others as side stories that will not happen.

The Same Message Cannot Be Multiplied Again

If Gu Ning had already set the B-satisfying historical three-quarters as the starting point, and then multiplied by B's ratio of 1.8, the new odds would go from three to 5.4, converting back to a probability of twenty-seven thirty-seconds. The computation can still yield a number between zero and one on paper, but the error lies in reusing information.

Because the starting odds of three to one already correspond to B being given, B is not a newly added observation. The follow-up should ask whether there is a new C and compare C within the model already conditioned on B; one cannot treat Tang Ke relaying the same message once more as another independent appearance.

This check shows that looking only at whether the result exceeds one is still insufficient. Erroneous multiplications can produce seemingly plausible high probabilities, and not all errors are as conspicuous as the opening 1.125. An update record must return to the information's provenance, not merely check the numerical range.

Chapter 11 will specifically trace how different sources share the original message. This chapter first addresses duplication within a model version: what the old judgment already used, and what new material actually adds, must both be stated before computation.

Which Information Was Already Given

The p, q, and r in the short formula all depend on a model background that is not repeatedly written out. Gu Ning knows the current task is at the stage of having sent the first version but without complete confirmation; she knows the window and uses a set of historical comparison classes. If the old probability also conditions on other significant information, the two message probabilities should be interpreted under the same background.

For example, as an application comparison: if the old judgment already targeted objects whose reception period was explicitly unavailable, one cannot take q and r from a historical group mixing all reception statuses and pretend the backgrounds are the same. Different inputs may each hold within their own scope, yet joined together they no longer correspond to the same judgment question.

The record can therefore first state the shared background, then the new observation B. The background need not encompass all real conditions, only let users know what this version retains. When the background must change, first check which inputs need re-estimation accordingly, rather than treating the original model's three numbers as parts movable anywhere.

This makes the prior more concrete. It is the old judgment before this observation, under the stated background, not a permanent property of R17 detached from all conditions. The posterior is likewise not a fixed label attached to reality, but the model's shares reorganized under the new observation.

Zero and One Are Not Ways to Express Strong Tone

If the old model gives A probability zero, then as long as the denominator of B remains positive, the A-and-B path remains zero, and routine updating cannot restore A to a positive share. If the old model gives A probability one, and B has positive probability under A, the update leaves it at one. Zero and one have strict meanings in the model—not merely "very unfavorable" or "especially confident."

If B has no probabilistic support at all under the old model, the denominator is zero, and a routine posterior cannot be obtained from the ratio formula. Gu Ning should re-check inputs, observational correspondence, and model scope, and identify whether reopening is needed, rather than filling zero over zero with fifty-fifty or temporarily changing the denominator to a very small number.

Finite material is usually insufficient to provide such strong grounds for future impossibility or necessity. If Gu Ning truly has a definitional exclusion condition, she may state it as such; but a person's strong tone cannot convert itself into endpoint probabilities, and a historical zero occurrence does not automatically become ontological impossibility.

The endpoint discussion is a model comparison and does not change the fact that the original eight items' p is five-eighths. It reminds readers that what updating can do is constrained by the original support's scope, and that reopening the model and converting within the original model are different actions. Allowing reopening is not a promise that a single multiplication will always restore the correct judgment later.

Displaying Decimals and Computational Precision Are Separate

Writing the original table's one-third as 0.333 is only an approximate display. If it is also truncated to 0.3 in computation, the denominator changes and the new result will not be exactly three-quarters. One cannot, to make the result match the old table, force it back to seventy-five percent after computing, without stating what precision was used in between.

This chapter's computation preserves the original fractions; the final display may write seventy-five percent or 75 percent as communication requires. Preserving precision aids re-checking; it does not mean an empirical judgment is exact to every displayed digit—the two notions must be stated separately.

The same holds when using computational tools. A tool can perform multiplication and division, but cannot know that p already includes B, nor discover on its own that the historical table omitted the pending reception time. Beyond checking numerical range and duplicated inputs, Gu Ning must still check objects and conditions; a machine not reporting an error cannot be taken as the application having been verified.

Outcome Probability and Input Estimation Are Not the Same Question

This chapter temporarily treats q and r as given inputs to compute the event's probability, without separately building a model to estimate these inputs' own uncertainty. Three out of five and one out of three suffice to define empirical frequencies, but do not prove that future probabilities of the same message exactly equal these ratios.

To study the unknown message probabilities further, one must separately state the estimation target, model, and additional assumptions. One cannot call the current event's three-quarters "q being three-quarters credible," nor call an input interval directly the interval of R17's outcome probability.

This division of labor prevents two layers of judgment from borrowing each other's name. Prediction asks whether R17 confirms within the target window; input evaluation asks how history and other material support p, q, and r; the two are related but are not the same event. The richer the mathematical representation, the more one must preserve whom exactly it assigns probability to.

When There Is Disagreement, First Compare Which Input Differs

If Tang Ke and Gu Ning both use this chapter's rules yet give different results, one need not immediately say one of them is more pessimistic or distrusts the client. One can first check side by side the target version, the old background, p, the message definition, q and r, and then look at the denominator and conversions.

If the target and all inputs are identical but the results differ, it may be an arithmetic or record problem; if the difference comes from different assumptions about r, then one should discuss why the message appears at its respective rate under not-A. Locating the dispute in the inputs reveals what material needs supplementing, rather than both sides repeatedly restating the final number.

It may also be that the targets look alike in name while one predicts receiving confirmation and the other predicts eventual completion of the repair. Then one should not average the two results but establish the objects separately. A shared updating rule does not let probabilities of different events be merged directly.

When the grounds cannot yet be unified, Gu Ning can preserve the two explicit scenarios and the location of disagreement, stating which this version adopts or temporarily does not adopt. Preserving disagreement is not treating all assumptions as equal weight, nor is it calibration because everyone has a number; it allows later material to correspond to and adjudicate the actual point of contention.

Updating Does Not Replace Outcome Adjudication

Even if the candidate model outputs three-quarters, whether the original event holds must still be checked within the window and the receipt conditions. A rising probability is not a confirmation having arrived; the recipient cannot replace the complete-acceptance-first-version original message with the update record.

If an explicit confirmation is obtained within the window later, the outcome record may write that the event holds, while preserving the earlier probability version made when it was unknown. If none is obtained, the non-holding does not directly show the computation was wrong; one must check the prediction group, input applicability, and possible fluctuations. Subsequent calibration handles precisely this relation; one cannot treat every non-holding as correct only if the probability had been zero.

At handoff, the computation page should also point to the original event card, the historical conditional table, and the original message text. With only the final seventy-five percent and none of these correspondences, the recipient cannot tell whether the same B was computed, nor distinguish scenario computation from actual adoption. If this version has not yet made a final prediction, its unadopted status is stated clearly; one does not fill in a number as a contemporaneous commitment to make the record look complete. A re-checkable record permits stating which step is not yet finished—this supports later observation better than retrospectively inventing an earlier judgment that never existed.

The Update Record Leaves Re-Checkable Changes

At 12:40 on Thursday, Gu Ning saves this chapter's computation record. The first layer is the restoration of the original eight-item empirical distribution, with p equal to five-eighths; the second layer is the conditional scenario if this coarse model is tentatively used as the current reference.

The record preserves the provenance of p, q, and r, the B scope, the two joint paths, the normalizing denominator, and the new condition's share, and notes that restoring the original table is not independent verification. The complete original utterance's pending reception time remains an application gap; R17 still has no complete confirmation; the task version and target window are unmodified.

The time points of old information are preserved in their original order: the 10:00 question, the 11:00 historical baseline, the 11:20 current message, the 12:00 conditional table and the later direction/role accounts. This round's computation was completed later and is not backfilled as Gu Ning having made the same probability earlier; the state of having had no numerical prediction then is also preserved.

The judgment this chapter yields is that updating requires connecting the prior share and the message likelihood within the same model, then normalizing by the observation's scope. A correct computation can reveal correspondences; it cannot prove the inputs true, the history transferable, or the complete message already handled.

The next question is no longer how to write the formula, but whether new material has truly added support. Different people reaching the same conclusion may share the same original message; more names cannot multiply the evidence indefinitely. Chapter 11 will trace the dependence along source paths, so that updating can continue without being pushed toward false conviction by repeated relay.