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Bayes Machine

A mechanical teaching simulator that stays faithful to the real mathematics by building an exact 100-case world, routing those cases through evidence gates, and then showing the update one line at a time. Turn the dials, make a prediction, run the machine, and watch the equation and the physical world match exactly.
World size: 100 exact cases
Gold chips: cases where the hidden condition is present
Red core: cases where the observation appears
Green glow: the chamber you are conditioning on
Prediction Ledger
Score
0
Streak
0
Latest result
Make a guess, then run the machine. The closer your guess is to the real updated probability, the more points you earn.
Control Cabinet
1. Set how many of the 100 cases truly have the hidden condition
This fixes the base rate. If you set 12, then P(H) = 12/100.
12
P(H)
12/100
P(not H)
88/100
2. Of the gold-chip cases, choose how many show the observation
This fixes how often the evidence appears when the condition is present.
9
P(E | H)
9/12
P(not E | H)
3/12
3. Of the steel-chip cases, choose how many also show the observation
This fixes how often the same evidence appears even when the condition is absent.
8
P(E | not H)
8/88
P(not E | not H)
80/88
4. Choose which observation you are conditioning on
You can condition on the evidence appearing or on it not appearing. Both are real Bayes updates.
5. Predict the updated probability before the machine reveals it
Your guess is scored against the exact posterior.
Your guess50%
6. Run the mechanism
Move step by step to see the world and the equation stay in sync.
Preset scenarios
These load exact counts, not rounded percentages.
Mechanical Chamber
Step
1 / 5
Whole world
P(H)
12/100
base rate
P(obs | H)
9/12
likelihood
P(obs | ¬H)
8/88
likelihood
Selected
—
pending
Posterior
—
reveal at step 5
GoalPredict the updated probability of the hidden condition after the machine sees the evidence.
WatchThe Posterior cell above and the Equation Ledger on the right. They must agree at step 5.
First movePress Next Step below to advance one stage. At step 5 you can Score guess.
1. Whole world
2. Split by H
3. Stamp E
4. Select chamber
5. Update
Whole world 100 exact cases before any sorting
H tray Gold chips: condition present
not H tray Steel chips: condition absent
H + E Gold chips showing the observation
not H + E Steel chips showing the same observation
Selected chamber Only the cases consistent with the chosen observation remain visible here
Hidden condition present
Hidden condition absent
Observation appears
Cases inside the conditioning chamber
Equation Ledger

Exact World Summary

Group Count Probability form Decimal

Bayes Step by Step

What this step means

Why the machine is mathematically faithful

Because the world contains exactly 100 constructed cases, every probability is an exact fraction of counts. The “weight” for the hidden condition is literally the number of selected gold chips divided by 100, and the final update is literally the fraction of selected chips that are gold. This makes the physical story and the algebra identical.
Short Game Loop
Mastery Notes
1) Move the first dial and watch how a small base rate can keep the updated probability low even when the evidence looks impressive. 2) Hold the gold tray fixed and raise the evidence among steel chips; the update will weaken because the observation is less distinctive. 3) Flip from conditioning on E to conditioning on not E and notice that “absence of evidence” is also a valid update. 4) Try to predict the posterior from the counts alone before opening the equation ledger.
Tip: once you are comfortable, ignore the decimals and reason entirely with counts. For example, if you condition on E, the updated probability is simply: selected gold chips / all selected chips. The full Bayes equation reduces to that exact ratio in this 100-case world.
Layered Equation Panel
Every tab shows the same update — Bayes' rule — from four different depths. Switch anytime.
Beat Script
The beats walk one hero quantity at a time. Step forward to advance the scripted path; the sim itself keeps all the controls available.