Post
by **Destination Goa** » Fri Feb 18, 2005 10:19 am

Adrian,

No, it's correct. If so, I would get the answer of 0.0 because even the very first step would be made from net profit >=1, but at the start we have only net profit of 0 available.

Ok, here is the math formula behind that code. I will use "i, j, k" for indices and "mm, kk, ll" for input values to stay synched with code variables.

a, b, c - number of wheel positions ( out of 28 ) where each symbol appears once, twice and three times respectively (this was discussed above). a=4, b=2, c=2.

v(i) - power of bet amount, i=0..nv-1

v(i) = 2^i.

2^(nv-1) <= ll, nv -> max

f(i, j, k) - probability of the state

i - number of rounds played (0 - played nothing, 1 - after 1st round)

j - bet index at this step (0..nv-1), actually Sten will bet for 2^j.

k - his overall profit. That is, what would be his ballance delta since the game begining if he leaves Ex at this point.

Now transitions:

For i<kk (i.e. for rounds when we don't care what is our profit), all states are propagated to 4 possible further states:

To f(i+1, (j+1)%nv, k-v[j]) with probability of f(i, j, k) * (28-a-b-c) / 28 // loss

To f(i+1, 0, k+v[j]) with probability of f(i, j, k) * a / 28 // win bet

To f(i+1, 0, k+v[j]*2) with probability of f(i, j, k) * b / 28. // win bet x 2

To f(i+1, 0, k+v[j]*3) with probability of f(i, j, k) * c / 28. // win bet x 3

So, as you may see, nothing is lost during processing i=0..kk-1. Sum(j=0..nv-1, k=-100..100) == 1.0 for any 0 <= i < kk.

Once i >= kk, we can propagate only states with positive net profit, i.e. states with k >= 1. You may see this in code as:

for(k = i<kk ? -BASE : 1 ; k<BASE ; k++)

Take a look at the initialization expression. If current 'i' (number of rounds was played) is less then 'kk', then we start at -INF, otherwise we start at 1. For sample input BASE is large enough to work as INF for us.

Also, in the end we amass only values >= 1.

The only difference in code from this statement is that array 'f' is shifted BASE elements on its 3rd argument to keep that index positive.

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