Randomness began at a gallop. The word comes from Old French randon — 'a rush, impetuosity, great speed' — from Frankish *rant, meaning 'a running'. In 14th-century English, 'at random' meant 'at full speed, without restraint'. A knight riding at random charged without controlling where his horse went.
The shift happened in three stages. First, 'at great speed' became 'without direction' — because an uncontrolled gallop goes wherever momentum takes it. Then 'without direction' became 'without purpose'. Finally, 'without purpose' became 'without pattern' — the mathematical sense that dominates today.
Each stage dropped one element of the original meaning and added another. Speed disappeared. Chaos remained. Probability entered.
The French cognate took a different path entirely. Randonnée — from the same root — slowed from 'gallop' to 'ramble' to 'hike'. A French randonnée is a long walk through the countryside. The word kept its sense of covering ground but lost its violence. English random kept the chaos but lost the motion.
The modern statistical meaning — 'each outcome equally likely' — was formalised in the 20th century. But the metaphor has not changed. A random number is one that has gone wherever momentum carried it, like a horse no one is steering. Randomness is the mathematical gallop.