A Question In Game Theory (i Think...) Sorry if this is the wrong place, but math is a kind of science.

[seec77 0]

I'm sorry if this is the wrong place for mathematical questions, but math is a scientific field.

Anyways, I have a question which I had with my friends over, and we can't seem to find the real answer ourselves, so I decided to ask you guys here, seeing as how the average IQ of this forum surpasses most typical people.

 

Anyways, it goes like this (and yes, this is taken from the Ludibrium Party Quest from Maple Story): Imagine a group of 9 switches. There is a unknown combination of 5 switches that need to be turned on for you to finish the level (or whatever your goal is). That is, if, for example, the combination is 13579, I have to turn on all those 5 switches for the door to open. Me and my friend were trying to think of how many possibilities there are for the secret combination, but neither of us being any game theory (is it even game theory?) experts, he started counting the different combinations (obviously stupid, as I tried to tell him ), and I tried calculate the factorial of 9 for up to 5 factors (9 x 8 x 7 x 6 x 5) which gave me a really large number. Both of these methods are obviously wrong, as mine is wrong because I also took into consideration the different options for ordering (i.e., I counted 13579 and 97531 as seperate options, even though they should not be such). So, I'm not only asking how many combinations there are, I'm also asking how I'm supposed to calculate such a thing. I tried binumeral distribution but I guess I just don't understand it well enough to apply it to this problem, or maybe it even has nothing to do with this problem!

 

Thank you very much, and again, sorry if this is not the best forum for my question!

Edited June 14, 2006 by seec77 (see edit history)

[yeh 0]

It's been a long time since I got anything to do with probability and statistics. But here is what I think. Or better still... if you already have the answer but don't know how to get it, post it here. Easier to figure out technique...I think it should be (9x8x7x6x5) / (5x4x3x2x1).

[Jeigh1405241495 0]

yeh is right. Its a fairly basic stats problem, you have 9 "objects" and must chosoe 5 of them, so its C(9,5) <--read '9 choose 5'Therefor its n!/(n-r)!r! or 9!/4!5! and since the 4! can cancel out some of the 9!... (9x8x7x6x5) / 5!It's weird, I hated stats but enjoyed this type of stuff in my combinatorics class. Either way, theres your answer

[lonebyrd 0]

Wow! I never got past algebra in school. You guys are amazing. I look at that and it looks like its written in alien! But seriously, I love math and wish I could do things like that because it fascinates me. I love it when people post these math problems.

[Jeigh1405241495 0]

I lovede math in HS, stopped loving it at the beginning of univeristy, now am starting to hit the odd course (like combinatorics) that are actually interesting and enjoyable math like this. Then again I think the prof I had deserves alot of the credit for my enjoyment of the combinatorics course buuuut, yea countring problems FTW! hahaEdit - If that's wrong now I am going to feel nice and stupid, WEEEHAW!

[seec77 0]

Thank you all so much! So it is something to do with binumeral distribution or whatever. I really gotta get me a book on these things. Does anyone have any recommendations of any free e-books/websites about these things where I can learn from scratch? And BTW, is this thing really related to game theory, or did I just think it is because it is taken from a game (MapleStory)?

 

Thanks again for the explanations!

[Arbitrary 0]

Hmmmm....I don't really think it's related to Game Theory (that's just combinations). Game Theory is sort of about desicion making (that's related to math, that is) under a difficult situation and get the most out of that situation. The most classic example is the one about the two criminals. They're taken to different jail cells. Then, they are both told that if they snitch on each other, they'll get 2 years in jail. If they work together, they'll get 6 months in jail. If one snitches and the other declares to work together, the snitcher will get out of jail free while the cooperative criminal will spend 2 years in jail. Game theory is then applied to a whole bunch of other desicion making theories....

 

I believe there was this one guy who came up with a strategy for winning game theory games. *BLEEP*-for tat (with forgiveness, optional). So basically if you're stuck in a game theory-like situation and you get to make a desicion more than once (i.e., not the jail one, because that's a one-time desicion) the first time you should play cooperative, in case the other person is also playing *BLEEP* for tat. Then, on your second turn, do whatever your opponent did last, and continue doing so for all turns thereafter. If there's a cycle of revenge, then that's where 'forgiveness' comes in--you cooperate to see if the other person will relent. That theory won an award somewhere...I believe....

 

It's quite fun, though.

 

Btw, here's some books about Game Theory...

 

Game Theory with economic applications by Bierman, H. S. and L. Fernandez(well, obviously it's about game theory's relations with economy...so it might or might not be of interest)

 

An Introduction to Game Theory by Martin J. Osborne

 

EDIT: Or...did you mean that you wanted books about combinatorics? I misread your post..Well...probably your math book should have some information on this stuff. The nCr, nPr, factorial section.

Edited June 26, 2006 by Arbitrary (see edit history)