You have a room full of 50 people and $2,500. Is it fair to distribute the money randomly amongst those 50 people? In one sense, it is "fair" given that everybody has the same chance of receiving a healthy payout — but you can guarantee equality by forgoing randomness. Just give each person $50. Simple.
Let's start again: there's a room of 50 people who each have $50. Every minute, each person will give a dollar to another random person in the room. Will the money be equally distributed amongst people or not?
That's a question posed by Uri Wilensky, a mathematician and professor at Northwestern University, and this interactive demo by Yaron Shemesh will help you arrive at the answer. (You can even adjust the number of people in the room, if you'd like).
Spoiler: the answer is "no." Giving everyone an equal start before introducing randomness essentially just delays the room's convergence towards a ridiculously unequal state. (The GIF above, which shows the wealth of each person in the room ranked from poorest to richest, was taken after only about 1,000 rounds of trading.) Now, because this system of allocation is totally random, eventually every person in the room will experience being the richest, the poorest and everywhere between — if each bar were a different color, you could watch Shemesh's simulation forever and track an individual as they hop around the distribution through each rank.
What will never change is the general shape of that distribution. With randomness determining who gets money in each round, periods of good luck and bad luck will shoot individuals up and down the rankings to positions they'll likely stay at for many rounds. You'll never see the distribution stabilize with everyone having roughly the same amount.
Of course, the economy isn't purely random, and this is as barebones a model as they come, but probability and luck still contribute to income inequality in the real world. If all the world's wealth were distributed randomly, the resulting imbalance would hardly be preferable to the kind we have anyway.