Short-Shot Cause Probabilities: From Bayes’ Theorem to Odds and Bayes Factors

In The Silent Factory, we traced the cause of short-shot defects. Between two suspects—temperature and pressure—the morning inspection strongly pointed to temperature. New evidence in the afternoon, however, shook that confidence.

There is a mathematical basis for that change in judgment.

This explanation follows the calculations in the story to show how Bayes’ theorem connects to the concise form of odds and Bayes factors. We will also check the numbers to see exactly what it means to update a judgment whenever new evidence arrives.

1. Distinguishing the numbers in the story

There are two causes under investigation in this case.

• $H_T$: Temperature is the cause of the short shots.
• $H_P$: Pressure is the cause of the short shots.

The prior probabilities drawn from historical records are:

Before seeing the new inspection results, the probability assigned to temperature was 60%, and to pressure, 40%.

The defect probabilities associated with each cause in the story are:

These are two different kinds of probability.

A judgment about the cause and the characteristics of the inspection results produced by that cause are different pieces of information. Bayes’ theorem connects them through actual observations.

2. Bayes’ theorem: probability after seeing evidence

Bayes’ theorem is:

The terms mean the following.

In this case, the probability of the evidence is:

For the observed data $D$, we weight its probability under temperature and under pressure by their respective prior probabilities, then add them.

Dividing by this denominator makes the posterior probabilities of the two hypotheses sum to 1, or 100%. This is called normalization.

3. How do inspection results become likelihoods?

The morning inspection found 5 defective products out of 50.

Each product is either defective or good. Coding a defect as 1 and a good product as 0 allows us to express the individual result with a Bernoulli distribution.

The number of defective products in these inspections is calculated using a binomial distribution.

• $p_H^k$: the contribution from $k$ defective products
• $(1-p_H)^{n-k}$: the contribution from $n-k$ good products
• $\binom{n}{k}$: the number of ways to place $k$ defects among the $n$ inspected products

The likelihoods of the morning result are therefore as follows.

The probability of finding exactly 5 defects out of 50 is approximately 16.29% under the temperature hypothesis and 3.46% under the pressure hypothesis.

The same evidence is much more naturally explained by the temperature hypothesis. This is what it means to compare causes through likelihoods.

4. Odds and Bayes factors: existing judgment and new evidence

Odds are the ratio of the probabilities of two hypotheses. Throughout this article, we calculate in the order temperature to pressure.

The prior odds are:

Before inspection, we considered temperature 1.5 times as likely as pressure to be the cause.

A Bayes factor, in contrast, is the ratio of the two likelihoods.

• $BF>1$: the new evidence favors temperature.
• $BF<1$: the new evidence favors pressure.
• $BF=1$: the new evidence does not change the relative assessment of the hypotheses.

While odds express the ratio in our existing judgment, the Bayes factor tells us how new evidence changes that ratio.

They are connected by this formula:

5. From Bayes’ theorem to the odds formula

Write Bayes’ theorem separately for temperature and pressure.

Dividing the two equations cancels the common term $P(D)$.

Separating the right-hand side into a likelihood ratio and a prior-probability ratio gives:

This is exactly “posterior odds = Bayes factor × prior odds.”

The probability and odds approaches express the same Bayes’ theorem in different forms. With the same data, either approach reaches the same result.

Both approaches also allow experience and records to inform the prior. The odds form is particularly useful for separating the roles of existing judgment and new evidence, and for presenting successive updates concisely.

6. Morning evidence: the probability of temperature rises to 87.6%

The morning inspection observed 5 defects out of 50. The Bayes factor is:

The identical combinatorial term $\binom{50}{5}$ cancels between numerator and denominator.

This evidence favors the temperature hypothesis approximately 4.714 times over the pressure hypothesis.

Multiply this by the prior odds of 1.5.

The temperature-to-pressure ratio is now approximately 7.071 to 1.

Converting the odds to the probability of the temperature hypothesis gives:

The probability of the temperature hypothesis rose from 60% to approximately 87.6%.

This explains why the engineer in the story came to suspect temperature so strongly.

Displayed values below are rounded for readability; subsequent calculations use unrounded values.

7. Does the probability approach give the same result?

Use the morning likelihoods calculated earlier.

The probability of the evidence is:

Substituting into Bayes’ theorem gives:

The result is again approximately 87.6%.

The probability approach calculates the denominator directly; the odds approach divides the two equations and cancels the common denominator. The calculation paths differ, but the mathematical content and result are the same.

8. Afternoon evidence: confidence is updated again

In the afternoon, 1 of the 50 newly inspected products was defective.

The Bayes factor for the afternoon evidence is:

This time it is less than 1. The afternoon evidence favors pressure over temperature.

We do not discard the morning evidence and start over. We incorporate the new afternoon evidence into the judgment reached by the morning.

Converting to the probability of the temperature hypothesis gives:

The probability of the pressure hypothesis is approximately 36.3%.

Temperature is still more likely. But 87.6% in the morning and 63.7% in the afternoon carry different weights in a decision. This is why the engineer stopped short of dismantling the heater and looked for more evidence.

The morning’s posterior becomes the afternoon’s prior. Adding new evidence to the judgment accumulated so far is sequential Bayesian updating.

Only the results of the 50 newly inspected products enter the afternoon update. The morning evidence is already included in the morning posterior and is not entered again.

9. Updating twice versus calculating once

Together, the morning and afternoon inspections contain 6 defects and 94 good products out of 100.

The Bayes factor for the combined evidence is:

This equals the product of the morning and afternoon Bayes factors.

The posterior odds are therefore identical.

The probability of the temperature hypothesis is again approximately 63.7318%.

In this case, collecting the evidence together or incorporating it as it arrives leads to the same conclusion. Sequential updating accepts new information without forgetting earlier evidence.

10. The value of Bayesian learning: following evidence toward the true cause

The initial 60% was a starting point. It rose to 87.6% in the morning and fell to 63.7% in the afternoon.

A falling probability does not mean learning has failed. It is the result of correcting a judgment in light of new evidence.

The strength of Bayesian updating is that we do not need to know the answer at the beginning. In this story, as actual inspection results accumulate and updating continues, the probability assigned to the true cause converges to 1.

Individual inspections can move the probability up or down because of sampling variation. But we do not discard earlier evidence whenever the number fluctuates. We retain the evidence accumulated so far and incorporate what is new.

Correcting an initially incomplete judgment with actual data and moving progressively toward the true cause: this is the value of Bayesian learning demonstrated by this case.

11. From calculations in a notebook to a system that drives operations

The story used a scene in which one person receives inspection results and calculates by hand to reveal the principles of Bayesian inference. It is an introductory case that lets readers directly follow the relationship among prior probability, likelihood, Bayes factor, and posterior probability.

To apply this principle continuously in real business operations, transaction records and work activities must be managed within a system. Orders and purchasing, production and inspection, inventory movements and shipments, equipment adjustments and staff actions must be connected: when they occurred, what they concerned, and under which conditions.

In actual short-shot management, “5 defects out of 50” is not enough on its own. We also need to know the LOT and equipment, the material and mold, the temperature and pressure at inspection, and any adjustments or maintenance in between. Those records let us examine the process conditions under which afternoon defects decreased and how that relates to actions already taken.

The value of systematization lies in connecting these records to judgment and execution.

Even when responsibility changes hands, the history of the incident, inspections, and actions remains accessible. There is less need to reconstruct earlier judgments from personal memory.

We can see which evidence arrived, how much it changed the judgment, and which action was chosen at the time. Review extends beyond the outcome to the decision process itself.

A quality issue can be examined together with production disruption, inventory shortages, and delivery impacts. Defect records from the shop floor become meaningful information for other departments’ plans and decisions.

Accumulating what was done and what changed afterward preserves experience and evidence as organizational assets for the next incident.

Not every record is treated as equivalent evidence. Duplicates and omissions must be checked, facts relevant to the problem selected, and related information managed so that it is not repeatedly counted as independent evidence. Reliable judgments require reliable records and appropriate reasoning together.

Actual actions also depend on more than the probability of a cause. Losses from equipment downtime, the risk of defective products escaping inspection, and the cost and time of additional inspection must also be considered. The system presents these grounds together, and the responsible person reviews operating conditions to decide on an action.

This connection is central to EXA’s pursuit of making business scientific.

This article introduced the basic principle of Bayesian updating within that cycle. In the stories that follow, the work and incidents of people using EXA systems will show, through concrete use cases, how transaction and activity records connect and support real decisions and execution.