6 ideas
Ancient peoples cast lots, knucklebones and dice to take a decision out of human hands, not to invoke chance. Because the outcome was read as the will of the gods, the device's apparent impartiality gave a verdict authority that no human judge could claim. This is why chance devices were used for dividing inheritances, choosing officials and assigning guilt.
A fair process can produce a sequence that looks patterned, such as six heads in a row, and a deterministic rule can produce output that looks disordered. Judging randomness by inspecting a single result therefore confuses appearance with generation.
When people try to produce or recognize random sequences, they switch outcomes too often and avoid long runs and clusters, because they expect every short stretch to look representative. The same belief drives the gambler's fallacy, the idea that a streak makes the opposite outcome 'due'. Independent trials have no memory, so genuine chance produces far more streaks than intuition allows.
In the 1970 U.S. Vietnam draft lottery, capsules holding birth dates were loaded month by month and then stirred too little before drawing. December birthdays, which went in last and sat near the top, received disproportionately low numbers and so had a higher chance of being drafted. A procedure that looked random on stage was measurably biased, and the method was redesigned the next year.
Casual shuffling of three or four riffles leaves exploitable structure from the previous deal. Adequate mixing takes a quantifiable amount of work, and people routinely do too little of it.
Generating randomness deliberately is difficult because physical devices carry bias and deterministic algorithms only imitate chance. Computer 'random' numbers are pseudorandom: they come from a formula, repeat eventually, and must be checked with statistical tests. This is why von Neumann said anyone using arithmetic to produce random digits is 'in a state of sin'. Seen this way, fair dice, shuffles and random-number tables are built and verified artifacts, not a free natural resource.