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Annoying maths question!

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Look Behind You
Space Ship Gamers inc.
EMERGENCY BROADCAST
#1 - 2011-11-29 00:01:52 UTC
If there are 10 football matches and 3 possible results for each how many different combinations of results could there be?

This is really confusing me after i got asked it today at work, ive been trying to work out a formula....am i sad?!

Any help from those maths geniuses would be great, its starting to grind my gears!
Liam Mirren
#2 - 2011-11-29 00:15:24 UTC
3^10 = 59049

Excellence is not a skill, it's an attitude.
Look Behind You
Space Ship Gamers inc.
EMERGENCY BROADCAST
#3 - 2011-11-29 00:46:35 UTC
I just realised how much of an idiot I am!
stoicfaux
#4 - 2011-11-29 01:40:41 UTC
Look Behind You wrote:
I just realised how much of an idiot I am!


And you needed help to get to that answer as well! Big smile

Pon Farr Memorial: once every 7 years, all the carebears in high-sec must PvP or they will be temp-banned.
CCP Shadow
C C P
C C P Alliance
#5 - 2011-11-29 03:40:32 UTC
Do you have other homework questions you'd like us to answer? Lol
.
Vicker Lahn'se
Republic Military School
Minmatar Republic
#6 - 2011-11-29 21:27:39 UTC
I really wish the forums would STOP EATING MY DAMN POSTS....

anyway..

Liam Mirren wrote:
3^10 = 59049


This is incorrect. This formula assumes that (win, loss, loss) is a different outcome from (loss, loss, win). You need to remove terms that are equivalent states. I feel stupid for not remembering the equation offhand, but I'll look it up in my thermodynamics textbook as soon as I get home.
stoicfaux
#7 - 2011-11-29 21:29:52 UTC  |  Edited by: stoicfaux
Vicker Lahn'se wrote:
I really wish the forums would STOP EATING MY DAMN POSTS....

anyway..

Liam Mirren wrote:
3^10 = 59049


This is incorrect. This formula assumes that (win, loss, loss) is a different outcome from (loss, loss, win). You need to remove terms that are equivalent states. I feel stupid for not remembering the equation offhand, but I'll look it up in my thermodynamics textbook as soon as I get home.


(win, loss, draw)? Each match has three states, and there are 10 matches. 3^10 gets my vote until someone redefines what a "match" means.

So... you're really saying that any combination of 5 wins and 5 losses are equivalent? Ex: (w, ..., w, l, ..., l) and (w, l, ..., w, l) are equivalent?

Permutations, combinatorics, dammit, where's my discrete math book?

edit: premature posting ftl.

Pon Farr Memorial: once every 7 years, all the carebears in high-sec must PvP or they will be temp-banned.
Vicker Lahn'se
Republic Military School
Minmatar Republic
#8 - 2011-11-29 21:34:34 UTC  |  Edited by: Vicker Lahn'se
stoicfaux wrote:
Vicker Lahn'se wrote:
I really wish the forums would STOP EATING MY DAMN POSTS....

anyway..

Liam Mirren wrote:
3^10 = 59049


This is incorrect. This formula assumes that (win, loss, loss) is a different outcome from (loss, loss, win). You need to remove terms that are equivalent states. I feel stupid for not remembering the equation offhand, but I'll look it up in my thermodynamics textbook as soon as I get home.


(win, loss, draw)?



Yes, I understand that there are three states for each individual game. The thing is, (win, loss, draw) is equivalent to (loss, win, draw). You need a factorial term to remove equivalent permutations.

Edit: Let's say you only play two games. There are a total of six possible outcomes:
2 win
2 loss
2 draw
1 win, 1 loss
1 win, 1 draw
1 loss, 1 draw

This is NOT 3^2.
Akita T
Caldari Navy Volunteer Task Force
#9 - 2011-11-29 21:45:16 UTC  |  Edited by: Akita T
Vicker Lahn'se wrote:
Yes, I understand that there are three states for each individual game. The thing is, (win, loss, draw) is equivalent to (loss, win, draw). You need a factorial term to remove equivalent permutations.

DUDE ! Ever bet on any sports events ?
You're not looking for "there were 4 wins, 2 draws and 4 losses", you're looking for "match 1 win, match 2 loss, match 3 draw, [...]" variations.
You need to specify each result for each individual match, matches are not interchangeable.
"Match 1 win, match 2 draw" is NOT the same as "Match 1 draw, match 2 win", even if in both cases you have "1 win, 1 draw" for 2 matches.


If matches would have just two possible states (win or loss) with no possible draws, "encoding" the results of 10 matches would be the same as encoding 10 bits, and the total number of possible outcomes would be 2^10=1024 variations.

With three possible states (draws also possible), you go into a 10-digit base-3 representation for 10 matches with 3 possible outcomes each, and the total number of variations that can happen is 3^10=59049.

If there would be 4 possible states to each match (win, loss, draw, canceled) you would have a 10-digit base-4 thing, with 4^10=1,048,576 variations.

[...]

If there would be 10 possible states (call them, oh, say, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) the representation would be a 10-digit base-10 thing, and you would have everything from 0000000000 to 9999999999 which so happens to be 10^10 variations, or 10000000000 variations.

Akita T evewiki profile | | | Build your own EVE-ready PC
stoicfaux
#10 - 2011-11-29 21:50:24 UTC  |  Edited by: stoicfaux
Combinations:
n = 3
r = 10
(n + r - 1)! / r!(n - 1)! =
(10+3-1)! / 10!(3-1)! =
12! / 10!2! =
12! / 10! * 1 / 2! =
12*11 * 1/2 =
66 combinations of wins, losses, and draws.

Disclaimer: Haven't done this stuff in decades. Probably shouldn't be doing math in public.

edit:
permutations: when order matters
combinations: when order doesn't matter

Pon Farr Memorial: once every 7 years, all the carebears in high-sec must PvP or they will be temp-banned.
Akita T
Caldari Navy Volunteer Task Force
#11 - 2011-11-29 22:08:51 UTC  |  Edited by: Akita T
Order has to matter.
Ever been to a bookie and said "but my ticket said 5 wins and 5 loses by the first team for the 10 matches today, why am I not getting a lot of money" when (some of) the teams you picked to win lost and (some of) those you picked to lose won but the totals were 5-5?
:P

Akita T evewiki profile | | | Build your own EVE-ready PC
Vicker Lahn'se
Republic Military School
Minmatar Republic
#12 - 2011-11-29 22:16:23 UTC
Akita T wrote:
Order has to matter.
Ever been to a bookie and said "but my ticket said 5 wins and 5 loses by the first team for the 10 matches today, why am I not getting a lot of money" when (some of) the teams you picked to win lost and (some of) those you picked to lose won but the totals were 5-5?
:P


But that's not what the OP asked for.
stoicfaux
#13 - 2011-11-29 22:16:47 UTC  |  Edited by: stoicfaux
Akita T wrote:
Order has to matter.
Ever been to a bookie and said "but my ticket said 5 wins and 5 loses by the first team for the 10 matches today, why am I not getting a lot of money" when (some of) the teams you picked to win lost and (some of) those you picked to lose won but the totals were 5-5?
:P


That's the "reasonable" assumption. IME, automatically assuming the "reasonable" assumption is unreasonable, where the "E" in "IME" equals *cough*large*telco*corporations*cough*

We could just be talking about a single team's record and just want to know how many total wins/losses/draws combinations are possible.

edit: tl;dr OP needs to clarify what they're asking for.

Pon Farr Memorial: once every 7 years, all the carebears in high-sec must PvP or they will be temp-banned.
Akita T
Caldari Navy Volunteer Task Force
#14 - 2011-11-29 22:34:03 UTC  |  Edited by: Akita T
Vicker Lahn'se wrote:
But that's not what the OP asked for.

That's not what you think the OP might have asked for, and you have a rather small chance (but still a chance) to be right.

stoicfaux wrote:
We could just be talking about a single team's record and just want to know how many total wins/losses/draws combinations are possible.

Hmm, I suppose that's also a (remote) possibility.


To both quotes above:

The way the OP phrased it makes it ALMOST certain that he was asking about the total number of variants you can fill a sports bet sheet with football match results. In that case, the result of each individual match matters, not the aggregate total of wins/losses by the first team or draws in each of the 10 matches.
The fact that he also quite rapidly accepted the first result with a borderline "duh!" reaction only cements the initial most likely assumption.

I suppose we could wait for the OP to clarify that to be absolutely certain, but I don't know if he's coming back around soon.


P.S. In the unlikely case that the OP really did mean the total aggregate number of results, then yes, it's 66, not 59049.
Easiest way to visualize it: imagine 3 bowls and 10 tokens. The bowls represent result type, the tokens represent number of results.
Start with all tokens in the first bowl. Move each token into bowl #2 one at a time, you get 10 moves, plus the initial variant, 11 possibilities. Move them all back into bowl #1, then put one into bowl #3, then out of the 9 remaining ones move each one at a time into bowl #2. Then put all of those in #2 back in #1 and move another into bowl #3.
Total possibilities : 11 + 10 + 9 + 8 + 7 + 6 + 5 + 6 + 4 + 3 + 2 + 1 = 66.

Akita T evewiki profile | | | Build your own EVE-ready PC
Astrid Stjerna
Sebiestor Tribe
#15 - 2011-12-01 03:36:15 UTC  |  Edited by: Astrid Stjerna
stoicfaux wrote:
Vicker Lahn'se wrote:
I really wish the forums would STOP EATING MY DAMN POSTS....

anyway..

Liam Mirren wrote:
3^10 = 59049


This is incorrect. This formula assumes that (win, loss, loss) is a different outcome from (loss, loss, win). You need to remove terms that are equivalent states. I feel stupid for not remembering the equation offhand, but I'll look it up in my thermodynamics textbook as soon as I get home.


(win, loss, draw)? Each match has three states, and there are 10 matches. 3^10 gets my vote until someone redefines what a "match" means.

So... you're really saying that any combination of 5 wins and 5 losses are equivalent? Ex: (w, ..., w, l, ..., l) and (w, l, ..., w, l) are equivalent?

Permutations, combinatorics, dammit, where's my discrete math book?

edit: premature posting ftl.


There are no 'equivalents' -- we're not looking at the results of the match, we're trying to determine the total number of possible permutations of win/loss/tie that can occur in ten matches.

I'll try and find some info....back later.

:Edit: Okay, here we go!

Akita T wrote:

Total possibilities : 11 + 10 + 9 + 8 + 7 + 6 + 5 + 6 + 4 + 3 + 2 + 1 = 66.


Close, but not quite. It's not adding, it's exponential multiplication. Each new game adds another three possibilities to the total.

3x9x27x81x242x729x2187x6561x19683x59049 = 177147.

(I do need a verification of the math, but I'm reasonably sure it's correct...)

I can't get rid of my darn signature!  Oh, wait....
Akita T
Caldari Navy Volunteer Task Force
#16 - 2011-12-01 05:10:44 UTC
Astrid Stjerna wrote:
Close, but not quite. It's not adding, it's exponential multiplication. Each new game adds another three possibilities to the total.
3x9x27x81x242x729x2187x6561x19683x59049 = 177147.
(I do need a verification of the math, but I'm reasonably sure it's correct...)

It's EITHER 66 in the unlikely case the OP meant "combination of total wins/loses/draws" OR (almost certainly) 3*3*3*3*3*3*3*3*3*3=3^10=59049 in case he meant total number of individual possibilities for each match in particular (as already mentioned and explained by most of the people in here).

It can't possibly be 3x9x27x81x242x729x2187x6561x19683x59049=3^(1+2+3+4+5+6+7+8+9+10)=3^55= 174,449,211,009,120,179,071,170,507 like your forumula suggest, nor is it possibly 3+9+27+81+242+729+2187+6561+19683+59049=88,571 which would be closer to the result you gave.
No idea where you got 177147 from anyway.

Akita T evewiki profile | | | Build your own EVE-ready PC
Selinate
#17 - 2011-12-01 06:05:35 UTC  |  Edited by: Selinate
Pretty sure the second post was the correct one...

If you have two games, the outcomes are (win,lose,draw),(win,lose draw).

Combinations are: (win,win), (win,lose), (lose,win), (lose,lose), (win,draw), (draw,win), (draw,draw), (lose,draw), (draw,lose)

that is 9 different combinations of outcomes. Extend it over 10 and you get 3^10.

EDIT: Maybe not. Questions like this usually depend on the teacher/text book. I would just go with whatever formula it uses given past example problems in using the same wording...
Lutz Major
Austriae Est Imperare Orbi Universo
#18 - 2011-12-01 08:19:35 UTC
Vicker Lahn'se wrote:

Edit: Let's say you only play two games. There are a total of six possible outcomes:
2 win
2 loss
2 draw
1 win, 1 loss
1 win, 1 draw
1 loss, 1 draw

This is NOT 3^2.

It's not 3^2, because you forgot :

1 loss, 1 win
1 draw, 1 win
1 draw, 1 loss

And I don't think the OPs question is ambiguous:
Look Behind You wrote:
If there are 10 football matches and 3 possible results for each how many different combinations of results could there be?
3 states, 10 games = 3^10. Period!


I think the confusing part for some is to abstract the win/loose/draw situation (and the fact, that the last time they used that kind of math was years ago): The question was not how often will team A win or loose. In fact there the question does not mention, whether two teams play 10 matches or 10 teams play 10 matches! The outcome is the important part and even the OP said nothing about win/loose/draw (although everybody assume this).
ChromeStriker
Sebiestor Tribe
Minmatar Republic
#19 - 2011-12-01 09:29:42 UTC
66 types of combination, 59049 combinations over 10 games

No Worries
Selinate
#20 - 2011-12-01 15:23:38 UTC
Lutz Major wrote:
Vicker Lahn'se wrote:

Edit: Let's say you only play two games. There are a total of six possible outcomes:
2 win
2 loss
2 draw
1 win, 1 loss
1 win, 1 draw
1 loss, 1 draw

This is NOT 3^2.

It's not 3^2, because you forgot :

1 loss, 1 win
1 draw, 1 win
1 draw, 1 loss

And I don't think the OPs question is ambiguous:
Look Behind You wrote:
If there are 10 football matches and 3 possible results for each how many different combinations of results could there be?
3 states, 10 games = 3^10. Period!


I think the confusing part for some is to abstract the win/loose/draw situation (and the fact, that the last time they used that kind of math was years ago): The question was not how often will team A win or loose. In fact there the question does not mention, whether two teams play 10 matches or 10 teams play 10 matches! The outcome is the important part and even the OP said nothing about win/loose/draw (although everybody assume this).


The word "combinations" can denote a completely different concept, though, where order doesn't matter, and hence (win,lose) is the same as (lose,win). It's ambiguous to anyone who's ever taken a statistics class...
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