OpenEuphoria

1. Gaussian Distribution Using Dice

• Posted by cklester <cklester at yahoo.com> Jul 30, 2004
• 498 views

What is the formula for calculating the odds of a number appearing
from a set of dice? I'm wanting to plot some bell curve graphs of this.
I have Googled and searched the forum/archive and nothing I can understand.

-=ck
"Programming in a state of EUPHORIA."
http://www.cklester.com/euphoria/

3. Re: Gaussian Distribution Using Dice

• Posted by Lewis Townsend <keroltarr at hotmail.com> Jul 31, 2004
• 472 views

ck,

Here is my solution but it doesn't do any graffing.
I think it is mathematically correct but it could be wrong.
I just wrote it real quick. It prints up something that
looks right to me.

include get.e

function sum( sequence s )
	atom a
	a = 0
	for i = 1 to length( s ) do
		a+= s[i]
	end for
	return a
end function

constant NumOfDice = 3
constant NumOfSides = 6

constant MaximumRoll = NumOfDice * NumOfSides

sequence chances, dice
chances = repeat( 0, MaximumRoll )
dice = repeat( 1, NumOfDice )

integer roll
for d = 1 to power( NumOfSides, NumOfDice ) do
	chances[sum(dice)] += 1
	dice[1]+=1
	for i = 1 to NumOfDice -1 do
		if dice[i] > NumOfSides then
			dice[i] = 1
			dice[i+1] += 1
		end if
	end for
end for


printf( 1, "The distobution of %d %d sided dice rolls:\n",
{NumOfDice,NumOfSides} )
puts( 1, "Value\tChance\n" )
for i = 1 to length( chances ) do
	printf(1, "%d\t%d\n", i&chances[i] )
end for
if wait_key() then end if

Below lies the signature of Lewis Townsend
Lewy T

5. Re: Gaussian Distribution Using Dice

• Posted by "Juergen Luethje" <j.lue at gmx.de> Jul 31, 2004
• 662 views

cklester wrote:

> This looks good, but, again, I don't understand the last math part...
>
>    http://mathforum.org/library/drmath/view/52207.html

This is not a Gaussian Distribution, BTW. A Gaussian Distribution is a
continuous distribution, whereas this is a discrete distribution.
However, the probabilities of sums on multiple dice *approach* a
Gaussian distribution as the number of dice is increased.

> I'd just like a FAST function like this:
>
>    aNumber = 13 -- what's the odds of the sum of 13
>    sides = 6 -- appearing when 6-sided dice
>    rolls = 4 -- are rolled 4 times and added together?
>
>    xFromY = instances( aNumber, sides, rolls )

[big snip]

On the URL mentioned above, the formula published 1937 by Uspensky is
provided. The formula uses the Binomial Coefficient (named C() there),
so we first need a function to calculate this. The Binomial Coefficient
is of fundamental meaning for combinatorial calculations, so it's
useful to have that function anyway.

global type nonnegative_int (object x)
   if integer(x) then
      return x >= 0
   end if
   return 0
end type

global type positive_int (object x)
   if integer(x) then
      return x > 0
   end if
   return 0
end type

global function binomial (integer n, nonnegative_int k)
   -- Binomial Coefficient ("n choose k"):
   -- Number of (unordered) subsets of k elements from a set of n elements.
   -- in : k: nonnegative integer
   --      n: integer >= k
   -- out: positive integer (not including 0)
   atom ret
   integer start

   if n < k then
      return -1              -- Illegal function call
   end if

   start = n - k
   if k > start then         -- swap k,start (so k is max.=floor(n/2))
      start = k
      k = n - k
   end if
   ret = 1
   for i = 1 to k do
      start += 1
      ret = ret*start/i
   end for

   return ret
end function

global function ways_to_roll_given_sum (positive_int x, positive_int n, integer
s)
   -- in : x: number of sides of each die
   --      n: number of dice
   --      s: given sum
   -- out: number of different ways to roll given sum with given dice
   integer ret

   ret = 0
   for k = 0 to floor((s-n)/x) do
      ret += power(-1,k) * binomial(n,k) * binomial(s-x*k-1,n-1)
   end for
   return ret
end function


-- Demo
integer Sides, Dice, Sum, NumberOfWays
atom Probability

Sides = 6
Dice = 4
Sum = 13

NumberOfWays = ways_to_roll_given_sum(Sides,Dice,Sum)
Probability  = NumberOfWays / power(Sides,Dice)
printf(1, "Number of ways(%d,%d,%d) = %d\n",   {Sides,Dice,Sum,NumberOfWays})
printf(1, "Probability   (%d,%d,%d) = %.2f\n", {Sides,Dice,Sum,Probability})

When you want to calculate the number of ways or probabilities for only
a few cases, this code might be faster than the code provided by Lewis.
Please check yourself.

Regards,
   Juergen

7. Re: Gaussian Distribution Using Dice

• Posted by "Juergen Luethje" <j.lue at gmx.de> Jul 31, 2004
• 491 views

Just 2 small comments (see below).

Lewis Townsend wrote:

> ck,
>
> Here is my solution but it doesn't do any graffing.
> I think it is mathematically correct but it could be wrong.
> I just wrote it real quick. It prints up something that
> looks right to me.
>
> }}}
<eucode>
> include get.e
>
> function sum( sequence s )
> 	atom a
> 	a = 0
> 	for i = 1 to length( s ) do
> 		a+= s[i]
> 	end for
> 	return a
> end function
>
> constant NumOfDice = 3
> constant NumOfSides = 6
>
> constant MaximumRoll = NumOfDice * NumOfSides
>
> sequence chances, dice
> chances = repeat( 0, MaximumRoll )
> dice = repeat( 1, NumOfDice )
>
> integer roll

Local variable roll is not used.

> for d = 1 to power( NumOfSides, NumOfDice ) do
> 	chances[sum(dice)] += 1
> 	dice[1]+=1
> 	for i = 1 to NumOfDice -1 do
> 		if dice[i] > NumOfSides then
> 			dice[i] = 1
> 			dice[i+1] += 1

                -- this makes it faster:
                else
                        exit

> 		end if
> 	end for
> end for
>
>
> printf( 1, "The distobution of %d %d sided dice rolls:\n",
> {NumOfDice,NumOfSides} )
> puts( 1, "Value\tChance\n" )
> for i = 1 to length( chances ) do
> 	printf(1, "%d\t%d\n", i&chances[i] )
> end for
> if wait_key() then end if
> </eucode>
{{{


Regards,
   Juergen

9. Re: Gaussian Distribution Using Dice

• Posted by Juergen Luethje <j.lue at gmx.de> Aug 03, 2004
• 488 views

cklester wrote:

> Juergen Luethje wrote:
>> 
>> cklester wrote:
>> 
>>> This looks good, but, again, I don't understand the last math part...
>>>
>>>    http://mathforum.org/library/drmath/view/52207.html
>> 
>> This is not a Gaussian Distribution, BTW. A Gaussian Distribution is a
>> continuous distribution, whereas this is a discrete distribution.
> 
> I'm glad we have geniuses on hand in the Euforum! :)

Me too. 
But that has nothing got to do with the difference between continuous and
discrete probability distributions. This difference is explained in any
statistical textbook on, say, page 23.

>> However, the probabilities of sums on multiple dice *approach* a
>> Gaussian distribution as the number of dice is increased.
> 
> Yeah, I knew that. :P

So the only remaining mystery is: Who wrote "Gaussian Distribution"
in the name of this thread. And why? 

>> When you want to calculate the number of ways or probabilities for only
>> a few cases, this code might be faster than the code provided by Lewis.
>> Please check yourself.
> 
> I haven't run any benchmarks on them yet, but your code works great!
> 
> Thank you!

Regards,
   Juergen