Random Values

Most algorithms you trace give the same output every time you run them with the same input. RANDOM breaks that pattern on purpose. RANDOM(a, b) is a command that returns a randomly selected integer from a to b, inclusive, where every integer in that range has an equal chance of being picked. A program that calls RANDOM is still correct even though its output changes run to run, because correctness here means the output always falls within the expected range and each value is equally likely, not that the output is fixed.

To find what RANDOM(a, b) can produce, list the integers from a up to b. The count of possible values is b - a + 1, and each one gets probability 1 divided by that count. So RANDOM(2, 6) can return 2, 3, 4, 5, or 6, five values total, each with probability 1/5. Run RANDOM(1, 6) a thousand times and graph the results as a bar chart: you won't get exactly 1000/6 for each bar, but with that many trials the bars end up roughly even in height, since equal likelihood is a statement about probability, not a guarantee of a perfectly even split in any particular batch of runs.

This idea connects directly to simulations, the topic that follows it. Simulations use repeated random choices to model real situations, like a weather model that picks rain or no rain each day, or a game that rolls dice for damage. None of that works unless RANDOM is trustworthy: unbiased, independent from call to call, and uniform across its range. Everything you know about equal likelihood here is what makes those larger simulations valid.