I am trying to figure out how to describe a betting game that involves multiple steps where at each step the probability of winning decreases but the payout increases. The player must start at step 1, and only wins a prize at step 4, 5, 6, or 7. Each step costs a set amount of money to play. The player cannot go backwards until step 4, 5, and 6 where the player returns to step 1 if he losses.
In addition, the player has the option to pay for increase chances of success for a set cost. At any step he can chose single step bonus (increase chance to move up one step), basic double step bonus (allow for possibility to move 2 steps), advanced double step bonus (increase chance to move up with possibility to move 2 steps).
How can I mathematically describe this problem so that I can determine:
- How much should each step cost to play if I want a positive expected value?
- How much should the payouts be for steps 4, 5, 6, and 7?
- How much should the player pay for increased odds of winning at each step?
If nothing else, I need to know the type of problem this is so I can figure it out on my own.