If something is done arbitrarily, it is decided or arranged without a real reason or a fair plan — often it feels unfair, as if someone just picked. A rule applied arbitrarily, a teacher who arbitrarily gives extra work: the point is “not following a clear, shared reason.” This is the adverb from arbitrary — it tells you how the deciding was done, not what the thing is. In careful talk it can also mean “in any way you like / picked at random,” without the angry sense of unfairness; maths and science use it more calmly (pick a number arbitrarily). In school English, the unfair reading is the one students meet first.