With unequal alternatives, decision time depends on information distribution, not count

What it is

For unequally likely alternatives, the uncertainty relevant to the Hick-Hyman law is not visible item count but the entropy of the choice-probability distribution. One almost-always selected alternative plus many rare ones can be quicker to decide than an equally sized uniform set. Count alone cannot describe decision burden.

Why it happens

People form priors from context and experience: high-probability responses can be prepared while rare responses need more discrimination. Entropy weights each probability by uncertainty; a more uniform distribution carries more average information to distinguish. Stable position, familiar labels, and cues can further alter subjective probability.

Studying it

Hold item count fixed while varying frequency distribution and randomising trials; compare time and error for uniform and skewed conditions, fitting entropy rather than count. Record whether users learned the distribution and control position and response mapping. Logs estimate frequency but task observation is needed to rule out low frequency caused by poor discoverability.

Where it stops holding

When users do not know the distribution, tasks switch suddenly, or rare actions have severe consequences, prior expectation should not safely govern design. Entropy describes selection, not label reading, visual search, or pointing. Mean efficiency cannot hide an undiscoverable rare critical task.

Related

• Same group: B1.13.2 When visual search dominates, time rises linearly with option count · B1.13.3 The law covers choice among known options, not time to understand them · B1.13.4 Fewer visible options can split one choice into layers without lowering total time · B1.13.5 Semantic similarity makes the law underestimate real decision time
• Nearby: B1.04 Hick-Hyman law · B1.07 Power law of practice
• Search terms: Hick-Hyman law · entropy · choice probability