Each item in the bank follows the two-parameter logistic (2PL) IRT model: the probability an examinee of ability θ answers item i correctly is
P_i(θ) = 1 / (1 + exp(-a_i (θ - b_i)))
b_i = difficulty (θ where P = 0.5)
a_i = discrimination (steepness of the curve)
After every response the engine re-estimates θ by Newton–Raphson maximum likelihood over every item administered so far (u_i = 1 correct, 0 incorrect):
score: L'(θ) = Σ a_i (u_i - P_i(θ))
information: I(θ) = Σ a_i² P_i(θ)(1 - P_i(θ))
update: θ ← θ + L'(θ) / I(θ) (step damped to ±1, θ clamped to ±4)
SE(θ̂) ≈ 1 / √I(θ̂)
The Max information strategy — real computerized adaptive testing (CAT), as used by the GRE and many licensing exams — administers whichever remaining item maximizes I(θ) at the current estimate, since that item narrows the confidence interval the fastest. The Random strategy administers items in random order for comparison: watch SE(θ̂) shrink far more slowly for the same number of items.
In the scene, the item bank sits on the difficulty (x) / discrimination (depth) plane; a glowing beam marks the running ability estimate. Grey spheres are unadministered (brightness ∝ information at θ̂ under the current strategy), green is a correct response, red is incorrect, and the rising path above the bank traces θ̂ after each step.