Hi all, this is my first post here! Forgive me if it's not in the right place.

I have a question that's been bugging me because I tend to be pretty good at math/statistics but I guess I'm a bit rusty.

Let's say you have $100 and the price to participate in the following game is $20:

You have a 5% chance to win $80
You have a 10% chance to win $60
You have a 15% chance to win $40
You have a 20% chance to win $20
You have a 50% chance to win $0

The expected value of your winnings will be 0.05*80+0.1*60+0.15*40+0.2*20 = $20 and therefore, given that the price to play is $20 as well, it is a zero-sum game. Suppose you are nevertheless interested in playing this game until you are bankrupt (lose the entire $100). My question is: what is the chance that you will be bankrupt in 10 trials or less? How do you calculate this? I suppose it's more the method of calculation I'm after, than the exact probabilities. 10 trials is just an example.

Yes the number of states is countably infinite (so as the dimension of the transition matrix). Being rigorous, you may need to know some technical regularities / defining a matrix in that way. Anyway, if you are just interested the game in finite time settings, you can always truncate the transition matrix, up to the maximum possible wealth within the given number of trials.