Revision 7T · Full Tutorial Edition · Updated 1 September 2026
PWR-127 · SELF GUIDED full tutorial
Update a probability on a fictional urn task one evidence card at a time
Belief updating is changing a prior confidence state after evidence. This lesson uses a fictional urn with coloured tokens and known evidence reliability. The learner writes a prior range and change rule, reveals one card at a time, records each update, checks alternatives and compares with a Bayesian benchmark—a calculation that combines the starting odds with how expected the evidence is under each option. It never targets real political, medical, identity or interpersonal beliefs.
What you will produceThe learner completes four fictional evidence sequences and makes directionally correct, proportionate updates that track the supplied likelihood information.
Full step-by-step individual tutorial · TLU-PWR-127
The learner completes four fictional evidence sequences and makes directionally correct, proportionate updates that track the supplied likelihood information.
Check: The update size is justified by supplied likelihoods.
Check cumulative coherence
After all cards, ask whether any evidence was double-counted or depended on an earlier card.
Why: Correlated evidence can create false certainty.
Check: Dependencies are marked and duplicates are not counted twice.
Plot the update path
Place the prior and every revised probability on one 0–100 line, annotate the evidence card and mark any all-or-nothing jump.
Why: A trajectory makes disproportionate or directionally inconsistent updates easier to inspect than the final estimate alone.
Check: Every movement has a card label, direction and magnitude, and unexplained jumps are corrected before benchmarking.
Compare and set reversal evidence
Calculate the Bayesian benchmark from the formula card: convert prior p to odds p/(1−p), multiply by P(evidence|A)/P(evidence|B), then convert the posterior odds O back with O/(1+O). Use only independent evidence. Explain any difference from the intuitive update and write what future card would move the belief back.
Why: A benchmark teaches the normative structure while preserving uncertainty and reversibility.
Check: Final estimate, benchmark, discrepancy and reversal condition are recorded.
4 · Worked example
See the whole method used once
Scenario
Urn A contains 80% blue tokens and Urn B 20%; one urn is selected with a 50% prior. One token is observed directly and replaced, so P(blue|A)=0.80 and P(blue|B)=0.20.
Walkthrough
Sal writes P(A)=50% before the first token-observation card.
He states a blue report should raise A, while a red report should lower it.
The first independent card is blue; he records random draw as an alternative and makes an intuitive update to 70%, not 100%.
A second blue card is only a duplicate record of that same token observation, so he marks it dependent and does not count it twice.
For the independent blue card he calculates prior odds 0.5/0.5=1, LR 0.80/0.20=4, posterior odds 4 and posterior P(A)=4/(1+4)=80%; he records under-updating and says an independent red card would lower the estimate.
Result
Sal updates in the correct direction, avoids double-counting and compares with a transparent benchmark. The task says nothing about his real-world beliefs.
5 · Right and wrong
Compare correct or safer execution with the common wrong version
Right and wrong comparison
Moment
Right / safer
Wrong / riskier
Why it matters
Prior
Write it before evidence.
Reconstruct a convenient prior afterward.
A hidden prior prevents evaluation of updating.
Evidence
Ask whether it is more likely under true or false.
Update because the evidence is vivid or repeated.
Vividness and repetition do not equal diagnosticity.
Dependence
Count a duplicated sensor report once.
Treat correlated copies as independent confirmation.
Double-counting creates false certainty.
Reversibility
Name future evidence that could move belief back.
Declare the final estimate permanently settled.
Honest beliefs remain conditional on evidence and model assumptions.
6 · Common mistakes
Spot the error and apply the correction
Common mistakes and corrections
Mistake
Fix
A range is collapsed to 0% or 100%.
Keep nonzero uncertainty unless the fixture logically rules an option out.
Source authority replaces likelihood.
Use the supplied reliability and relevance for the specific card.
Repeated cards are assumed independent.
Trace shared source or measurement before multiplying evidence.
The final benchmark is copied without explaining discrepancy.
Name under-update, over-update, arithmetic error or different assumption.
7 · Practice
Turn the steps into a usable skill
First session
Set one fictional prior.
Predeclare evidence directions.
Reveal three cards sequentially.
Check dependence and compare with benchmark.
Write reversal evidence.
Repeat plan
Complete one new fictional sequence twice a week for two weeks. Vary prior and reliability, then include one non-diagnostic and one dependent card. Do not substitute real controversial claims.
Progress when
Update direction is correct for every card.
Magnitude responds to supplied diagnosticity.
Dependent evidence is not double-counted.
Do not progress when
Content becomes real, identity-linked or consequential.
The learner feels pressure to conform to a source rather than evaluate evidence.
Likelihoods or ground truth are unavailable.
8 · Check the result
Measure what changed
Directional and proportional accuracy of probability updates on fictional sequences.
How: For each independent card compute prior odds p/(1−p), likelihood ratio P(E|A)/P(E|B), posterior odds and posterior probability O/(1+O). Record intuitive estimate, benchmark, absolute difference, duplicate-evidence handling and reversal condition for each sequence.
Good result: Update direction matches the supplied likelihood ratio, the benchmark calculation reproduces from the formula card, dependent evidence is excluded and any intuitive–benchmark gap is explained and retained. No ten-percentage-point cutoff is treated as general open-mindedness.
This does not prove: It does not show real-world open-mindedness, correct values, resistance to manipulation or authority over another person’s beliefs.
Self-check
Was the prior recorded before evidence?
Is this card more expected under true or false?
Is it independent of prior cards?
What future evidence would reverse the update?
9 · Stop, adapt or get help
Keep the safety boundary practical
Stop and get help
Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
Stop if the exercise is used to pressure agreement or collect sensitive beliefs.
Use accountable domain evidence and legitimate decision processes for consequential real-world updating.
Accessibility and adaptations
Use probability bars, frequencies out of 100 or tactile tokens instead of formulas.
Read cards aloud one at a time with a scribe.
Allow calculator support and extra time while preserving reveal order.
10 · Evidence and limits
Why these instructions are here
primary research
Fact-checking experiments across four countries improved accuracy for targeted claims, but effects belong to the tested claims and contexts.
A ten-country vaccine-misinformation correction produced only a small accuracy change and did not change attitudes or intentions, limiting broad transfer claims.
Progress is saved only in this browser on this device.
Step-by-step learner mode
Each activity includes its success check, a nearby accessible alternative and an “I’m stuck” correction path. Alternatives preserve the target where possible; when they change the task, Titan labels them as related rather than equivalent.
01
Write the prior range
Record the starting probability as a point or honest range and cite the supplied base rate.
Why this step exists
A visible prior prevents hindsight and shows how far evidence moved belief.
Success check
The prior and reason are fixed before card one.
I’m stuck on this step
Reset: Re-read this authored instruction — “Record the starting probability as a point or honest range and cite the supplied base rate.” — and its success check, then attempt only this step.
Possible snag: A range is collapsed to 0% or 100%.
Correction: Keep nonzero uncertainty unless the fixture logically rules an option out.
Stop / get help: Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
02
Translate the prior into counts
Represent the starting probability as expected cases out of 100, split between the two urn options. Keep the original percentage beside it.
Why this step exists
Concrete counts make later evidence changes visible without requiring formula fluency.
Success check
The frequency table totals 100 and matches the declared prior before any evidence card opens.
I’m stuck on this step
Reset: Re-read this authored instruction — “Represent the starting probability as expected cases out of 100, split between the two urn options. Keep the original percentage beside it.” — and its success check, then attempt only this step.
Possible snag: The result from “Represent the starting probability as expected cases out of 100, split between the two urn options. Keep the original percentage beside it.” does not yet meet this declared check: The frequency table totals 100 and matches the declared prior before any evidence card opens.
Correction: Return to the start of “Translate the prior into counts”, reduce complexity or pace, and repeat only the part needed to satisfy: “The frequency table totals 100 and matches the declared prior before any evidence card opens.”
Stop / get help: Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
03
Predeclare update direction
For each possible evidence type, state whether it would be more expected if the claim were true, false or equally expected.
Why this step exists
Likelihood direction matters more than whether evidence sounds impressive.
Success check
Each card type has an expected direction before reveal.
I’m stuck on this step
Reset: Re-read this authored instruction — “For each possible evidence type, state whether it would be more expected if the claim were true, false or equally expected.” — and its success check, then attempt only this step.
Possible snag: The result from “For each possible evidence type, state whether it would be more expected if the claim were true, false or equally expected.” does not yet meet this declared check: Each card type has an expected direction before reveal.
Correction: Return to the start of “Predeclare update direction”, reduce complexity or pace, and repeat only the part needed to satisfy: “Each card type has an expected direction before reveal.”
Stop / get help: Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
04
Reveal one card
Open only the next card, record its reliability and alternative way it could arise, then update once.
Why this step exists
Sequential logging prevents the final story from erasing intermediate reasoning.
Success check
The log shows card, reliability, alternative and new probability.
I’m stuck on this step
Reset: Re-read this authored instruction — “Open only the next card, record its reliability and alternative way it could arise, then update once.” — and its success check, then attempt only this step.
Possible snag: Source authority replaces likelihood.
Correction: Use the supplied reliability and relevance for the specific card.
Stop / get help: Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
05
Match update to diagnosticity
Move more for evidence with a larger true-versus-false likelihood difference and little for weak or equally likely evidence.
The update size is justified by supplied likelihoods.
I’m stuck on this step
Reset: Re-read this authored instruction — “Move more for evidence with a larger true-versus-false likelihood difference and little for weak or equally likely evidence.” — and its success check, then attempt only this step.
Possible snag: The result from “Move more for evidence with a larger true-versus-false likelihood difference and little for weak or equally likely evidence.” does not yet meet this declared check: The update size is justified by supplied likelihoods.
Correction: Return to the start of “Match update to diagnosticity”, reduce complexity or pace, and repeat only the part needed to satisfy: “The update size is justified by supplied likelihoods.”
Stop / get help: Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
06
Check cumulative coherence
After all cards, ask whether any evidence was double-counted or depended on an earlier card.
Why this step exists
Correlated evidence can create false certainty.
Success check
Dependencies are marked and duplicates are not counted twice.
I’m stuck on this step
Reset: Re-read this authored instruction — “After all cards, ask whether any evidence was double-counted or depended on an earlier card.” — and its success check, then attempt only this step.
Possible snag: Repeated cards are assumed independent.
Correction: Trace shared source or measurement before multiplying evidence.
Stop / get help: Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
07
Plot the update path
Place the prior and every revised probability on one 0–100 line, annotate the evidence card and mark any all-or-nothing jump.
Why this step exists
A trajectory makes disproportionate or directionally inconsistent updates easier to inspect than the final estimate alone.
Success check
Every movement has a card label, direction and magnitude, and unexplained jumps are corrected before benchmarking.
I’m stuck on this step
Reset: Re-read this authored instruction — “Place the prior and every revised probability on one 0–100 line, annotate the evidence card and mark any all-or-nothing jump.” — and its success check, then attempt only this step.
Possible snag: The result from “Place the prior and every revised probability on one 0–100 line, annotate the evidence card and mark any all-or-nothing jump.” does not yet meet this declared check: Every movement has a card label, direction and magnitude, and unexplained jumps are corrected before benchmarking.
Correction: Return to the start of “Plot the update path”, reduce complexity or pace, and repeat only the part needed to satisfy: “Every movement has a card label, direction and magnitude, and unexplained jumps are corrected before benchmarking.”
Stop / get help: Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
08
Compare and set reversal evidence
Calculate the Bayesian benchmark from the formula card: convert prior p to odds p/(1−p), multiply by P(evidence|A)/P(evidence|B), then convert the posterior odds O back with O/(1+O). Use only independent evidence. Explain any difference from the intuitive update and write what future card would move the belief back.
Why this step exists
A benchmark teaches the normative structure while preserving uncertainty and reversibility.
Success check
Final estimate, benchmark, discrepancy and reversal condition are recorded.
I’m stuck on this step
Reset: Re-read this authored instruction — “Calculate the Bayesian benchmark from the formula card: convert prior p to odds p/(1−p), multiply by P(evidence|A)/P(evidence|B), then convert the posterior odds O back with O/(1+O). Use only independent evidence. Explain any difference from the intuitive update and write what future card would move the belief back.” — and its success check, then attempt only this step.
Possible snag: The final benchmark is copied without explaining discrepancy.
Correction: Name under-update, over-update, arithmetic error or different assumption.
Stop / get help: Stop if the fixture shifts to a real political, clinical, identity, relationship or legal claim.
Correct versus incorrect execution
These accessible process diagrams are built from the tutorial’s own right/wrong teaching. They are not anatomical illustrations and do not add technique beyond the canonical tutorial.
Prior — A hidden prior prevents evaluation of updating.
Correct / safer
Write it before evidence.
Wrong / riskier
Reconstruct a convenient prior afterward.
Evidence — Vividness and repetition do not equal diagnosticity.
Correct / safer
Ask whether it is more likely under true or false.
Treat correlated copies as independent confirmation.
Reversibility — Honest beliefs remain conditional on evidence and model assumptions.
Correct / safer
Name future evidence that could move belief back.
Wrong / riskier
Declare the final estimate permanently settled.
Method-structure checklist
10 of 10 structural checks present
✓ Ordered, Power-specific instructions — present
✓ Every activity has a success check — present
✓ Materials or supplied records are declared — present
✓ Measurement or assessment rule is present — present
✓ Tutorial-specific troubleshooting is present — present
✓ Stopping or escalation boundary is present — present
✓ Every activity has an adjacent alternative — present
✓ Correct-versus-incorrect comparison is present — present
✓ Evidence context is bound to the Power record — present
✓ Planning metadata is present — present
The method-readiness band and presence checklist assess tutorial presentation and are separate from evidence quality for the underlying Power. They are automated editorial aids, not human approval.
Manual editorial sign-off: Pending. This tutorial must not display a human-approved state until an identified editor signs the exact content hash.