OT: Home Sick for the Holidays, or Probability and Yahtzee

Krsqk · #8 December 25th, 2003, 06:46 PM

Quote:

Originally posted by Imperator Fyron:
I forget if you want to use permutations or combinations here... it will be one over one of the following formulae:

n is the total number of items, r is the number of those items you want.

Combination (order does not matter):
n! / [ ( n - r )! * r! ]

Permutation (order matters):
n! / (n - r)!

So for probability, you have either

[ ( n - r )! * r! ] / n!

or

(n - r)! / n!

Depending on which is the one you want.

These formulae work when r=1, but don't seem to work when r>1.

code:

[ ( 6 - 1 )! * 1! ] / 6!

[ 5! * 1 ] / 720

[ 120 * 1 ] / 720

120 / 720

1/6, or 16 1/6%

but,

code:

[ ( 6 - 5 }! * 5! ] / 6!

[ 1! * 120 ] / 720

120 / 720

1/6, or 16 1/6% (should be 83 1/3%)

Obviously, the odds that I will roll a 6 and the odds that I will roll less than 6 are not both 1/6 (at least on a standard d6

).

It's worse with the permutation formula:

code:

( 6 - 1 )! / 6!

5! / 720

120 / 720

1/6, or 16 1/6%

but,

code:

( 6 - 5 )! / 6!

1! / 720

1/720, or ~.1389%

Things are even more exciting when you factor in more dice, such as the odds to roll at least one six with five d6:

code:

[ ( 30 - 5 }! * 5! ] / 30!

[ 25! * 120 ] / 2.6525285981219105863630848e+32

[ 15,511,210,043,330,985,984,000,000 * 120 ] / 2.6525285981219105863630848e+32

1,861,345,205,199,718,318,080,000,000 / 2.6525285981219105863630848e+32

7.0172483965587413863275932241449e-6

.0000070172483965587413863275932241449, or ~.0007%

I'd have thought my chances of rolling a six would have been slightly better than that! That's pretty close to 1/6^4.

Offhand, I'd guess there's a problem with the denominator in this formula--could it be n!/r, which would make the formula:

code:

r [ ( n - r )! * r! ] / n!

5 [ ( 6 - 5 }! * 5! ] / 6!

5 [ 1! * 120 ] / 720

5 [ 120 ] / 720

600 / 720

5/6, or 83 1/3%

but, then again:

code:

3 [ ( 6 - 3 )! * 3! ] / 6!

3 [ 3! * 3! ] / 720

3 [ 6 * 6 ] / 720

3 * 36 / 720

108 / 720

3/20, or 15% (should be 50%)

Any insight here?

[ December 25, 2003, 16:48: Message edited by: Krsqk ]