How to calculate probability

Column a is a list of possible results that can be generated by the use of 4
six sided dice together ranging from 4-24. What I want to calculate in
column B is the number of combinations that the dice can produce that would
total to any given result.

How do I do this? Can I do this?

Assuming fair dice

4 can occur only if each die shows a 1: prob=1/6^4

5 can occur if 1 die shows 2 and the rest show 1: prob=COMBIN(4,1)/6^4

6 can occur if 1 die shows 3 and the rest show 1: prob=COMBIN(4,1)/6^4
or if 2 dice show 2 and the rest show 1: prob=COMBIN(4,2)/6^4
(add the two probabilities)

7 can occur if 1 die shows 4 and the rest show 1: prob=COMBIN(4,1)/6^4
or if 1 die shows 3, 1 die shows 2 and the rest show 1:
prob=MULTINOMIAL(1,1,2)/6^4
or if 3 dice show 2 and 1 die shows 1: prob=COMBIN(4,1)/6^4

....

Jerry

Al wrote:

> Column a is a list of possible results that can be generated by the use of 4
> six sided dice together ranging from 4-24. What I want to calculate in
> column B is the number of combinations that the dice can produce that would
> total to any given result.
>
> How do I do this? Can I do this?
>

So how do I turn all of that into something reasonably neat use in my
caculations for column B?

"Jerry W. Lewis" wrote:

> Assuming fair dice
>
> 4 can occur only if each die shows a 1: prob=1/6^4
>
> 5 can occur if 1 die shows 2 and the rest show 1: prob=COMBIN(4,1)/6^4
>
> 6 can occur if 1 die shows 3 and the rest show 1: prob=COMBIN(4,1)/6^4
> or if 2 dice show 2 and the rest show 1: prob=COMBIN(4,2)/6^4
> (add the two probabilities)
>
> 7 can occur if 1 die shows 4 and the rest show 1: prob=COMBIN(4,1)/6^4
> or if 1 die shows 3, 1 die shows 2 and the rest show 1:
> prob=MULTINOMIAL(1,1,2)/6^4
> or if 3 dice show 2 and 1 die shows 1: prob=COMBIN(4,1)/6^4
>
> ....
>
> Jerry
>
> Al wrote:
>
> > Column a is a list of possible results that can be generated by the use of 4
> > six sided dice together ranging from 4-24. What I want to calculate in
> > column B is the number of combinations that the dice can produce that would
> > total to any given result.
> >
> > How do I do this? Can I do this?
> >
>
>

Becase of the multiple modes by which a total can be obtained, this doesn't
lend itself to a simple expression (unlike the 2-dice problem). If I wanted
to calculate the results in a hurry without thinking about the combinatorics,
then I would use helper columns as follows:

In C1:F1 enter the number 1
In C2 enter the formula =IF(SUM(D2:$F2)=COUNT(D2:$F2),IF(C1=6,1,C1+1),C1)
Copy C2 and Paste into D2:E2
In F2 enter the formula =IF(F1=6,1,F1+1)
Copy C2:F2 and Paste into C3:F1296
In G2 enter the formula =SUM(C1:F1)
Copy G2 and Paste in to G3:G1296

Now the probability of getting a total of n across the 4 dice can be
calculated as
=COUNTIF($G$1:$G1296,n)/6^4

Jerry

"Al" wrote:

> So how do I turn all of that into something reasonably neat use in my
> caculations for column B?
>
> "Jerry W. Lewis" wrote:
>
> > Assuming fair dice
> >
> > 4 can occur only if each die shows a 1: prob=1/6^4
> >
> > 5 can occur if 1 die shows 2 and the rest show 1: prob=COMBIN(4,1)/6^4
> >
> > 6 can occur if 1 die shows 3 and the rest show 1: prob=COMBIN(4,1)/6^4
> > or if 2 dice show 2 and the rest show 1: prob=COMBIN(4,2)/6^4
> > (add the two probabilities)
> >
> > 7 can occur if 1 die shows 4 and the rest show 1: prob=COMBIN(4,1)/6^4
> > or if 1 die shows 3, 1 die shows 2 and the rest show 1:
> > prob=MULTINOMIAL(1,1,2)/6^4
> > or if 3 dice show 2 and 1 die shows 1: prob=COMBIN(4,1)/6^4
> >
> > ....
> >
> > Jerry
> >
> > Al wrote:
> >
> > > Column a is a list of possible results that can be generated by the use of 4
> > > six sided dice together ranging from 4-24. What I want to calculate in
> > > column B is the number of combinations that the dice can produce that would
> > > total to any given result.
> > >
> > > How do I do this? Can I do this?

It is less obvious than my previous suggestion, but you could also count the
number of ways to get a given total from n dice (numerators to divide by 6^n
for the probability) from the numerators for rolling n-1 dice.

In A1:A6 enter the number 1 -- these are the numerators for rolling 1 die
In B1 enter the formula =SUM(OFFSET(A1,MAX(-5,1-ROW()),0,MIN(6,ROW()),1))
Copy B1 and paste into B2:B11 -- these are the numerators for rolling 2 dice
Copy B1 and paste into C1:C16 -- these are the numerators for rolling 3 dice
...

this implements the formula given by
http://www.research.att.com/cgi-bin/...i?Anum=A063260

Jerry

"Jerry W. Lewis" wrote:

> Becase of the multiple modes by which a total can be obtained, this doesn't
> lend itself to a simple expression (unlike the 2-dice problem). If I wanted
> to calculate the results in a hurry without thinking about the combinatorics,
> then I would use helper columns as follows:
>
> In C1:F1 enter the number 1
> In C2 enter the formula =IF(SUM(D2:$F2)=COUNT(D2:$F2),IF(C1=6,1,C1+1),C1)
> Copy C2 and Paste into D2:E2
> In F2 enter the formula =IF(F1=6,1,F1+1)
> Copy C2:F2 and Paste into C3:F1296
> In G2 enter the formula =SUM(C1:F1)
> Copy G2 and Paste in to G3:G1296
>
> Now the probability of getting a total of n across the 4 dice can be
> calculated as
> =COUNTIF($G$1:$G1296,n)/6^4
>
> Jerry
>
> "Al" wrote:
>
> > So how do I turn all of that into something reasonably neat use in my
> > caculations for column B?
> >
> > "Jerry W. Lewis" wrote:
> >
> > > Assuming fair dice
> > >
> > > 4 can occur only if each die shows a 1: prob=1/6^4
> > >
> > > 5 can occur if 1 die shows 2 and the rest show 1: prob=COMBIN(4,1)/6^4
> > >
> > > 6 can occur if 1 die shows 3 and the rest show 1: prob=COMBIN(4,1)/6^4
> > > or if 2 dice show 2 and the rest show 1: prob=COMBIN(4,2)/6^4
> > > (add the two probabilities)
> > >
> > > 7 can occur if 1 die shows 4 and the rest show 1: prob=COMBIN(4,1)/6^4
> > > or if 1 die shows 3, 1 die shows 2 and the rest show 1:
> > > prob=MULTINOMIAL(1,1,2)/6^4
> > > or if 3 dice show 2 and 1 die shows 1: prob=COMBIN(4,1)/6^4
> > >
> > > ....
> > >
> > > Jerry
> > >
> > > Al wrote:
> > >
> > > > Column a is a list of possible results that can be generated by the use of 4
> > > > six sided dice together ranging from 4-24. What I want to calculate in
> > > > column B is the number of combinations that the dice can produce that would
> > > > total to any given result.
> > > >
> > > > How do I do this? Can I do this?