Imagine there is a bowl with 10 marbles of which 4 are blue, 3 are green, 2 are red, and 1 is yellow.
You now may pick a random marble and put it back into the bowl. If you picked a blue marble, you may pick again with a chance of $p_{blue} = 0.5$. If you picked a green marble, you may pick again with a chance of $p_{green} = 0.25$. If it was a red marble, the chance to pick again is $p_{red} = 0.1$. And lastly, if it was a yellow marble, you may not pick another marble.
Now, what I generally want to know is: What is the probability that a marble of a certain color has been picked $k$ times?
Let's say we want to know with what probability a red marble has been picked exactly 1 time. What is the probability?
I struggle to even write down this probability function. Maybe Markov chains can help here?