Explanation 1
Prior: Probability before the new signal.
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Lesson 13 of 16
Update a prior probability after observing a signal and explain the base-rate effect.
Theory
Update a prior probability after observing a signal and explain the base-rate effect.
Explanation 1
Prior: Probability before the new signal.
Explanation 2
Likelihood: Probability of the signal under a stated condition.
Explanation 3
Posterior: Updated probability after the signal.
Terminology
Probability before the new signal.
Baseline quality-issue rate.
Probability of the signal under a stated condition.
Alert given an issue.
Updated probability after the signal.
Issue given an alert.
Notation and formulas
P(B given A) = P(A given B)P(B) / [P(A given B)P(B) + P(A given not B)P(not B)]
Define every event before substituting values, then validate the result and denominator.
Interactive Mission
P(event given positive)
0.3191
True-positive contribution / all positive contributions
| Actual group | Positive | Negative |
|---|---|---|
| Event | 360 | 40 |
| No event | 768 | 8832 |
A rare base rate can leave many false positives even when sensitivity is high.
Worked example
Scenario
An automated alert is sensitive, but confirmed issues are rare.
Sensitivity is not positive predictive probability.
R connection
Run and edit the code, inspect intermediate output, and use the movable R Code Guide while practising.
Live R Lab
Update a prior event probability using sensitivity and false-positive probability.
Ready to run
Dataset provenance
Original synthetic data generated for STATLAB Academy. No textbook data used.
Interpretation safeguard
A highly accurate signal can still have a low positive predictive value when the event is rare.
Reflection
Name the event, denominator, probability interpretation, and one operational limitation.
Assistant
Help me reason through Bayes' Theorem: From Prior to Posterior Probability by asking questions about my event definitions, denominator, formula choice, assumptions, and interpretation. Do not give a quiz answer before I submit.
Exit check
Use a hint before submitting if needed. Full calculations and interpretations appear after submission.
Checkpoint
Question 1 of 2. Answered 0/2. Passing score: 70%.
Resource
Explore additional probability explanations and exercises from properly licensed or publisher resources.
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