I've got a set of sample data and I'm looking to see if it's possible to generalize a binomial formula to give a closed form solution to this. If not, would it be possible to write a program to do this? Apologies in advance if it isn't really clear, but I just have the data and I'm trying to build the rules around it.
Background: Basically all time periods are split in two, for example two days of the week, Monday and Friday. So, The first time period is Monday, second is Friday, third is Monday, fourth is Friday etc...
Each day an event occurs, say a coin flip where Heads is a win and Tails is a loss. There are different payoff systems depending on what day the coin is flipped.
On Monday the coin is flipped.
If the coin wins (heads) on a Monday, there is a payoff of M1 (one day starting on monday).
The coin is flipped again on Friday, if it wins again, there is a payoff of M2 (two days in a row, starting on) but if it loses there is a payoff of F1 (one day starting on friday.
So each time the coin loses, it is reset to 1 of the current day (M1 or F1) and the series starts again from there.
Total payoff is the sum of payoffs at each period.
I'm trying to think of the best way to work with this, but as an example of they way I've been trying it, the possible outcomes for a 5 period game with 4 coin tosses (or is this a 4 period game...) that starts on Monday are:
Payoffs when no wins (4 Choose 0 solutions):
(M1 + F1 + M1 + F1 + M1)
Payoffs with one win (4 Choose 1 solutions):
(M1 + F1 + M1 + F1 + F2), (M1 + F1 + M1 + M2 + M1), (M1 + F1 + F2 + F1 + M1), (M1 + M2 + M1 + F1 + M1)
Payoffs with two wins (4 Choose 2 solutions):
(M1 + F1 + M1 + M2 + M3), (M1 + F1 + F2 + F3 + M1), (M1 + M2 + M3 + F1 + M1), (M1 + M2 + M1 + M2 + M1), (M1 + F1 + F2 + F1 + F2), (M1 + M2 + M1 + F1 + F2)
Payoffs with three wins (4 Choose 3 solutions):
(M1 + M2 + M3 + M4 + M1), (M1 + F1 + F2 + F3 + F4), (M1 + M2 + M3 + F1 + F2), (M1 + M2 + M1 + M2 + M3)
Payoffs with four wins (4 Choose 4 solutions):
(M1 + M2 + M3 + M4 + M5)
So that's it... It seems a bit confusing so I might not have explained it well, but will answer any questions. If it's hard to understand I can make a Tree image that might make more sense, let me know.