For the Monte Carlo Method stuff in particular[1], I get the sense that the most iconic "Hello, World" example is using MC to calculate the value of pi. I can't explain it in detail from memory, but it's approximately something like this:

Define a square of some known size (1x1 should be fine, I think)

Inscribe a circle inside the square

Generate random points inside the square

Look at how many fall inside the square but not the circle, versus the ones that do fall in the circle.

From that, using what you know about the area of the square and circle respectively, the ratio of "inside square but not in circle" and "inside circle" points can be used to set up an equation for the value of pi.

Somebody who's more familiar with this than me can probably fix the details I got wrong, but I think that's the general spirit of it.

For Markov Chains in general, the only thing that jumps to mind for me is generating text for old school IRC bots. :-)

[1]: which is probably not the point of this essay. For for muddying the waters, I have both concepts kinda 'top of mind' in my head right now after watching the Veritasium video.