Was that a real flicker, or just the eye's own static? Every detection is two overlapping bells — noise alone, and signal-plus-noise — and one movable line where you decide to say 'yes'. Drag the criterion, slide the signal strength, and trace the trade-off between catching the signal and crying wolf.
Signal detection theory: two equal-variance Gaussians — noise at 0, signal-plus-noise at d′ — and a criterion you drag. Hit rate = 1 − Φ(c − d′), false-alarm rate = 1 − Φ(c), where Φ is the normal CDF (computed live from an erf approximation). Sliding d′ separates the bells (more sensitivity); moving the criterion trades hits against false alarms and sets the bias — the ROC point traces the classic curve. All exact. The key insight is real: sensitivity and bias are separable; you can be accurate yet trigger-happy, or cautious yet blind.
Equal-variance Gaussian noise is the textbook SDT model; real neural/perceptual noise can be unequal-variance or non-Gaussian, which bends the ROC. The formulas here (d′, criterion, hit and false-alarm rates, the ROC) are computed exactly for that classic model — the figure is that a real detector's internal distributions are exactly two matched bells.