what you describe, in general, falls within the: "drawing random samples from a random distribution (RD)". Except that you additionally want to favour certain outcomes more. That can be achieved by transforming your RD. e.g. skew it, squash it, lift the tails, etc., prior to drawing.

Drawing samples from an RD requires you to build the cumulative RD and then land randomly on any 'y'. Corresponding 'x' will be the random sample you need. You have a discrete RD, a histogram. So, build a cumulative histogram, transform the histogram to give better odds to unpopular items and then draw the sample.

I would keep the 2 tasks separately.

3day edit: in the 2nd paragraph swapped 'x' and 'y'. So basically, draw a random number from 0 to 1 this will be the 'y', the spot on the y-axis. The corresponding 'x' will be a random sample wrt this RD.

bw, bliako