The binomical or in this case 3 probabilities instead 2 is a better alternativeomid_dt wrote:Assuming normal distribution may not be totally accurate, but it is widely used in many fields (and has resulted in some horrible results, e.g., the economic model of LTCM hedge fund, which lost billions).
But anyway, in our case, is there a better alternative?
that you can use at least for n that is small(and it seems to me possible to get exact value at least for number of games that is not bigger than 200
and I believe that even for 1000 games it is possible if it does not need to be done in one second and you are ready to wait some hours).
The main problem to do it is that you need to have probability with hundreds of digits(or even thousands of digits in case of 1000)
I do not know how to calculate numbers like 200!/(100!*50!*50!) in the C language and also calculating numbers like 0.4^100*0.31^50*0.29^50 when you need accurate number may be a problem but if you solve this problem then you may only need to calculate sum of a million probabilities like 0.4^(200-W-D)*0.31^D*0.29^W for
all the values that W+2D=0,W+2D=1,W+2D=2,....so you get probability distribution of W+2D and based on probability distribution you can find better confidence interval.
I think that the main technical problem is to write function to calculate all of this.
I do not know how much time every calculation takes and I assume that you may need more than regular calculation on integers but I believe that something in the order of million calculation is practically not a problem(it may be a problem if n is in the order of thousands).
I know that people tend to assume that the normal distribution is good enough for big N*P but I think that it is always better to get exact value and not approximated value if getting an exact value is possible.
Uri