Update formulas — compact

This chapter is a quick reference. Derivation lives in chapter 5; interactive drill — chapter 7.

Posterior — Bayes’ rule

After each attempt first compute the posterior — updated belief given the observation (correct / incorrect):

$$P(L \mid \text{correct}) = \frac{P(L) \cdot (1 - P(S))}{P(L) \cdot (1 - P(S)) + (1 - P(L)) \cdot P(G)}$$ $$P(L \mid \text{wrong}) = \frac{P(L) \cdot P(S)}{P(L) \cdot P(S) + (1 - P(L)) \cdot (1 - P(G))}$$

Learning step

Then add the chance of learning during this attempt:

$$P(L_{\text{new}}) = P_{\text{posterior}} + (1 - P_{\text{posterior}}) \cdot P(T)$$

With $P(T) = 0.1$ the learner always gains a little — even on mistakes.

P(solve) — for prediction

When choosing the next task we need the probability the student solves this problem now:

$$P(\text{solve}) = P(L) \cdot (1 - P(S)) + (1 - P(L)) \cdot P(G)$$

“Knows and doesn’t slip” plus “doesn’t know but guesses.”

Implementation

These three formulas are the adaptive engine’s heart. In code they fit in ~15 lines:

export function pSolve(pL: number, params: BktParams = DEFAULT_BKT): number {
  return pL * (1 - params.pSlip) + (1 - pL) * params.pGuess;
}

export function bktUpdate(
  pL: number,
  observedCorrect: boolean,
  params: BktParams = DEFAULT_BKT
): number {
  const { pSlip, pGuess, pTransit } = params;
  const posterior = observedCorrect
    ? (pL * (1 - pSlip)) / (pL * (1 - pSlip) + (1 - pL) * pGuess)
    : (pL * pSlip) / (pL * pSlip + (1 - pL) * (1 - pGuess));
  return posterior + (1 - posterior) * pTransit;
}

Formula cheat sheet

What	Formula	Role
Update (correct)	$\frac{P(L)(1-P(S))}{P(L)(1-P(S)) + (1-P(L))P(G)}$	“correct answer → more confidence”
Update (wrong)	$\frac{P(L)P(S)}{P(L)P(S) + (1-P(L))(1-P(G))}$	“mistake → less confidence (no panic)”
Learning	$P_{\text{post}} + (1-P_{\text{post}}) \cdot P(T)$	“can learn during attempt”
Predict	$P(L)(1-P(S)) + (1-P(L))P(G)$	“probability of solving?”

All four are deterministic. No sampling, no black boxes, no neural nets — eighth-grade arithmetic.