I am writing a program meant to simulate a gambling game. I want to be able to set the total amount of possible winnings for each day. But I do not know the number of players I will have that given day.
Say I have 3 possible prizes: $\$1$, $\$5$ and $\$10$ and I want to give out a maximum of $100\$$ every day. I can have between 500 and 10000 players in any given day.
How do I configure the probability of winning each prize, so that a player playing early in the morning has approximatively the same change of winning as a player who plays in the evening?