A group of people have 100,000 beans, with an indeterminate amount of beans per person.
If someone gives me 4 beans, I give them back $1.
If someone gives me 3 beans, there is an X% chance I give them back $1.
If someone gives me 2 beans, there is a Y% chance I give them back $1.
If someone gives me 1 bean, there is a Z% chance I give them back $1.
The 4-bean chance is 100% and X > Y > Z.
They can repeat this as many times as they want but once they give me the beans I do not give them back, meaning the two opposite extremes would be I give away $25,000 in total to people who all give me exactly 4 beans or I give away $100,000 in total to people who all give me exactly 1 bean and got very lucky.
How do I calculate and/or visualize the chances that I give out over $50000 given a random sample of bean amounts given to me? How do I calculate/visualize this given a specified amount of beans given to me for certain #s of beans$^1$?
How do I calculate/visualize the probability that I give out $75,000?
$^1$If I know that 10,000 people will give me exactly 4 beans, how do I calculate the odds for the remaining 60,000 beans? Assuming the chances of people giving 3 beans, 2 beans, and 1 bean are equal.
I understand there is some permutation/combinatorial probability and game theory involved, but I don't really even know where to start in terms of thinking about and framing a formula for the problem.