Update a probability without losing the base rate

Try

A positive quality screen sounds persuasive, but its meaning depends on how common the defect was before screening. In an original synthetic model,10% of items are defective. The screen is positive for80% of defective items and20% of nondefective items. Predict whether most positive items must be defective. Do not confuse the80% sensitivity with the requested reverse conditional.

Predict

Imagine1000 items rather than manipulating symbols first. Predict the sizes of the defective and nondefective groups, then the positives from each. A fairly small false-positive rate can contribute many positives when the nondefective group is large. Keep the two branches separate until the final conditioning step.

Experiment

The expected-frequency ledger is:

Actual condition	Items	Positive	Negative
Defective	100	80	20
Nondefective	900	180	720
Total	1000	260	740

These are model-scaled frequencies, not results of an actual experiment. Among the260 positives,80 are defective, so P(defective given positive)=80/260=4/13, about30.8%. The answer is below one half despite80% sensitivity. Draw the two positive branches merging into one group and label which part belongs in the numerator.

Understand

Bayes’ rule reverses conditioning by relating it to a joint probability. Let D mean defective and + mean positive. The numerator P(+ given D)P(D) is the probability of both D and +. The denominator P(+) adds the positive branches across the exhaustive, disjoint possibilities D and not D. Thus P(D given +)=P(+ given D)P(D)/P(+), provided P(+)>0.

Here P(+)=0.8·0.1+0.2·0.9=0.26. Dividing the true-positive joint probability0.08 by0.26 gives4/13. Sensitivity is P(+ given D); the false-positive rate is P(+ given not D). Specificity is the complement of that latter rate,0.8 here. The equal numeric values in this example do not make these concepts interchangeable.

The prior or base rate P(D) belongs to the target population before this evidence. The posterior P(D given +) describes the conditioned group. Holding the two nondegenerate conditional screen rates fixed, increasing prevalence increases the fraction of positives that are true positives. A posterior from one factory or population is therefore not a universal property of the screen.

Transporting a model requires more than changing one number casually. Conditional performance can also vary with defect severity, equipment, thresholds or procedures. Document where each rate came from and whether the new setting supports it. A mathematically correct update can still be unreliable when its inputs are poorly estimated or mismatched.

Bayes calculations do not themselves choose an action. Inspection costs, consequences of missed defects and available follow-up tests affect decisions. Those considerations should be added explicitly rather than disguising a decision threshold as a probability fact. This unit uses quality-control scenarios to practice the bookkeeping, not to prescribe real diagnostic or operational decisions.

Apply

Rebuild the ledger with prevalence20% while keeping the stated screen rates. Per1000 items,160 defective and160 nondefective items test positive, giving posterior1/2. Explain why sensitivity did not change but the posterior did. In the factory practice task, use production shares as the prior and condition on all defects at the end.

Continue

Next assign numbers to outcomes and average them using probabilities. This leads to expectations, variances and counts of repeated successes.