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// Explanation of Terms

A = Advance to Referenced Level

R = Replay Referenced Level

D = Drop to Referenced Level

C = Special Credit (Game Ends)

E = Elimination (Game Ends)

W = Win Game

# = Level Reference

// Game Structure

LEVEL 1 = A2,A2,R1,C1,E,E,E,E,E

LEVEL 2 = A3,A3,R2,R2,D1,E,E,E,E

LEVEL 3 = A4,A4,R3,R3,R3,E,E,E,E

LEVEL 4 = A5,A5,R4,D3,D3,D2,E,E,E

LEVEL 5 = A6,A6,R5,R5,D4,D4,E,E,E

LEVEL 6 = W,C6,C6,E,E,E,E,E,E

I am trying to determine the probability of advancing from one level to the next or winning the game at Level 6, on a level by level basis. For Levels 1, 3, and 6, I am able to do these calculations quickly and easily because the total number of permutations is small. However, the number of possible outcomes increases dramatically at Levels 2, 4, and 5, because of the chances of dropping one or two levels as part of the game sequence.

Where replays are concerned in these calculations, I am wanting to go at least 10 degrees deep. So, for instance, take Level 3 for example, the calculation of probability to advance to Level 4 would look like this:

(A4)+(R3^1*A4)+(R3^2*A4)+(R3^3*A4)+(R3^4*A4)+(R3^5*A4)+(R3^6*A4)+(R3^7*A4)+(R3^8*A4)+(R3^9*A4)+(R3^10*A4)

So, I am wanting to calculate (mathematically or through simulation) the probability of advancing from:

LEVEL 2 TO LEVEL 3

LEVEL 4 TO LEVEL 5

LEVEL 5 TO LEVEL 6

Because the system described above is static, the parameters to advance from one level to the next are definable. I have those definitions, but do not know what software program is best for this or necessarily how to program into such. I am not completely ignorant when it comes to programming and believe that I could probably pick it up without a lot of difficulty, but a little guidance or links to similar references to what I am trying to do would be extremely helpful.

Thanks in advance, Brian.

Coincidentally, my co-worker and I just programmed a relatively elegant program for the probability of drawing x red balls from a bag of y balls (having two colors). Gets kind of fun when you get into it.

Coincidentally, my co-worker and I just programmed a relatively elegant program for the probability of drawing x red balls from a bag of y balls (having two colors). Gets kind of fun when you get into it.

Isn't that just a hypergeometric distribution? Or do you not know how many of each type there are in the bag?

Good point! We set it up so that we wouldn't know how many of each type. Though thinking back, it would have been much easier if I had used that distribution. We derived the equations ourselves in the program.