Align: In practice is actually does still matter. Consider the 3 possibilities. You chose a nothing box, He uncovers a nothing box. You change You win You chose the other nothing box, He uncovers a nothing box. You change You win You chose the correct box, he uncovers a nothings box, You change You lose.
That's not quite how it goes. Let's say you chose box A initially, he opened box C (which was empty). At the point in the game where you have to make your choice whether to switch or not, there are two boxes in the game (A and B). Either is equally likely to contain the keys. That you chose a box previously doesn't matter, since you can now switch. The choice is between two boxes, each with an equal chance of containing the keys. That's similar to chosing heads or tails. The coin has an equal chance of landing on either side, so no matter which one you pick, your chances of winning are 50%. Whether you look at it mathematically or practically, the choice is the same: There is no choice. You're at the whim of fate, and can't do anything to increase the odds. DiscoZombie got it right the first time.
AbraWould you kindlyJoin Date: 2003-08-17Member: 19870Members
<!--quoteo(post=1679581:date=May 27 2008, 10:17 PM:name=Xyth)--><div class='quotetop'>QUOTE(Xyth @ May 27 2008, 10:17 PM) <a href="index.php?act=findpost&pid=1679581"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->So there is an airplane on a treadmill...
Align: In practice is actually does still matter. Consider the 3 possibilities. You chose a nothing box, He uncovers a nothing box. You change You win You chose the other nothing box, He uncovers a nothing box. You change You win You chose the correct box, he uncovers a nothings box, You change You lose.
If you change you have a 2/3 in chance.<!--QuoteEnd--></div><!--QuoteEEnd-->
This would only be valid if you had any idea of more than one box. Practically-No-Go as we say in quizspeak.
Xyth's example is correct. Initially, you have a 66% chance to choose an empty box (three to choose from, we assume that you have no idea where the keys are, and therefore choose at random). Since an empty box is removed before "round two" and it is never the one you selected, this means that if you initially selected an empty box, the OTHER remaining box is certain to be the one holding the keys. On the other hand, if you initially selected the box holding the keys, the other box will of course be empty.
So we now have two possible scenarios: You initially choose the key box, or you initially choose an empty box. Since there are two empty boxes and one key box, you are twice as likely to choose an empty box as you are to choose the key box (meaning you have a 66% chance of choosing an empty box). Since round two will always consist of one empty box and the key box, and you have a 66% likelihood of initially having chosen an empty box, there is now a 66% likelihood that the OTHER box will be the one holding the keys. Therefore, in order to maximise your chances at winning, you should switch.
<!--quoteo(post=1679585:date=May 27 2008, 05:10 PM:name=Abra)--><div class='quotetop'>QUOTE(Abra @ May 27 2008, 05:10 PM) <a href="index.php?act=findpost&pid=1679585"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->This would only be valid if you had any idea of more than one box. Practically-No-Go as we say in quizspeak.<!--QuoteEnd--></div><!--QuoteEEnd--> Can you explain your reasoning behind this?
This isn't a complicated problem, it's high-school statistics (Which just so happens to be where I heard it first). Write out all the possibilities and the answer is clear...
High school statistics? I heard this question on my first year in college! <img src="style_emoticons/<#EMO_DIR#>/nerd-fix.gif" style="vertical-align:middle" emoid="::nerdy::" border="0" alt="nerd-fix.gif" />
Well, some parts did, such as customization (one of my favorites *sob*)
Off-Topic is alive and kicking though <img src="style_emoticons/<#EMO_DIR#>/biggrin-fix.gif" style="vertical-align:middle" emoid=":D" border="0" alt="biggrin-fix.gif" /><!--QuoteEnd--></div><!--QuoteEEnd-->
Well, some parts did, such as customization (one of my favorites *sob*)
Off-Topic is alive and kicking though <img src="style_emoticons/<#EMO_DIR#>/biggrin-fix.gif" style="vertical-align:middle" emoid=":D" border="0" alt="biggrin-fix.gif" /><!--QuoteEnd--></div><!--QuoteEEnd--> The NS2 general discussion and I&S are also quite alive. I spend most of my time on the board these days trying to get people to think through on their suggestions; supporting great new concepts or changes to existing designs while pointing out the flaws in awful ideas.
If you really have problems getting it, turn it into an n problem.
You got n boxes. So the chance that the box you choose contains the keys are 1/n. The chance that the key is in any of the unchoosen boxes is (n-1)/n Quizmaster opens m boxes. Which gives us: (n-1)/(n*(n-m-1)) for the chance of one of the unchoosen, unrevealed boxes containing the key.
Please remember, that m:[1;n-2]
Now that we know the mathematics behind this (which I just rethought of. I did not write them off somewhere (though I did this before in grade 13)) All we need to compare is the chance for th initial box and the chance for the potential trade box.
Assumption: (n-1)/(n*(n-m-1)))>1/n
(n-1)/(nn-nm-n)>1/n
(nn-n)/(nn-nm-n)>1
Thats of course always the case, because n and m are >0 by definition. But just in case that you dont see it: nn-n=x -> x/(x-nm)>1 Should be clear now.
As you can see it is always worth changing. You can advance the problem, by allowing more than 1 box to be choosen, but then the result is not surprising, cause depending on how many boxes are opened, you either be better off with changing or staying.
<!--quoteo(post=1679581:date=May 27 2008, 10:17 PM:name=Xyth)--><div class='quotetop'>QUOTE(Xyth @ May 27 2008, 10:17 PM) <a href="index.php?act=findpost&pid=1679581"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->So there is an airplane on a treadmill...
Align: In practice is actually does still matter. Consider the 3 possibilities. You chose a nothing box, He uncovers a nothing box. You change You win You chose the other nothing box, He uncovers a nothing box. You change You win You chose the correct box, he uncovers a nothings box, You change You lose.
If you change you have a 2/3 in chance.<!--QuoteEnd--></div><!--QuoteEEnd-->That's the mathematical way, yes. Not the "common sense" way, which tells us that we have no idea which box contains the key so it doesn't matter how or when or which we pick.
<!--quoteo(post=1679615:date=May 28 2008, 11:47 AM:name=Align)--><div class='quotetop'>QUOTE(Align @ May 28 2008, 11:47 AM) <a href="index.php?act=findpost&pid=1679615"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->That's the mathematical way, yes. Not the "common sense" way, which tells us that we have no idea which box contains the key so it doesn't matter how or when or which we pick.<!--QuoteEnd--></div><!--QuoteEEnd--> "Common sense" in this case is wrong though. You don't know which box contains the key, but that doesn't mean that it doesn't matter which one you pick, as the keys are not equally likely to be in either box. Common sense is to pick the box with higher odds, namely the one you DIDN'T pick initially. What you describe may be common thinking for those who don't statistically analyse the problem, but their reasoning is flawed: They argue that they cannot make an informed decision because they lack information, but fail to realise that they CAN make an <u>educated guess</u> based on the information that IS available.
<!--quoteo(post=1679615:date=May 28 2008, 05:47 AM:name=Align)--><div class='quotetop'>QUOTE(Align @ May 28 2008, 05:47 AM) <a href="index.php?act=findpost&pid=1679615"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->That's the mathematical way, yes. Not the "common sense" way, which tells us that we have no idea which box contains the key so it doesn't matter how or when or which we pick.<!--QuoteEnd--></div><!--QuoteEEnd-->
It might not be the "common sense way" but that does not remove the practical nature of the solution. It increases your chances of getting what you want so you should do it. Very practical.
ShockehIf a packet drops on the web and nobody's near to see it...Join Date: 2002-11-19Member: 9336NS1 Playtester, Forum Moderators, Constellation
The NS community isn't dead. A lot of us are still great friends, chat a lot, etc. We just generally don't play as much (if any) NS as we did, primarily because, well, it's <i>old</i>, and we've done it to death.
NS2, I'm sure we'll all be back moaning/commenting (delete to taste) on the game. <img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
<!--quoteo(post=1679643:date=May 28 2008, 02:55 PM:name=Shockwave)--><div class='quotetop'>QUOTE(Shockwave @ May 28 2008, 02:55 PM) <a href="index.php?act=findpost&pid=1679643"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->The NS community isn't dead. A lot of us are still great friends, chat a lot, etc. We just generally don't play as much (if any) NS as we did, primarily because, well, it's <i>old</i>, and we've done it to death.
NS2, I'm sure we'll all be back moaning/commenting (delete to taste) on the game. <img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
- Shockwave<!--QuoteEnd--></div><!--QuoteEEnd-->
*sends you a hug data packet*
Yeah pretty much all my attention is on helping in any way I can with making NS2 a positively kickass game. And that's with "spare time", which really just means "procrastinated time" on account of my generally slammed schedule.
Like I said before, when the community thins like this it's clear who the long-term members are that have true loyalty to the game and the community really are. The hosts of the party are the one's left when the drunks leave.
<!--quoteo(post=1679711:date=May 29 2008, 01:55 PM:name=lolfighter)--><div class='quotetop'>QUOTE(lolfighter @ May 29 2008, 01:55 PM) <a href="index.php?act=findpost&pid=1679711"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->Mnah. Just because I live in <country> it doesn't mean I'm a patriot.<!--QuoteEnd--></div><!--QuoteEEnd-->
<!--quoteo(post=1679914:date=Jun 1 2008, 01:07 AM:name=DrSuredeath)--><div class='quotetop'>QUOTE(DrSuredeath @ Jun 1 2008, 01:07 AM) <a href="index.php?act=findpost&pid=1679914"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->To be fair, only Americans make fun of Canada, while the rest of world makes fun of US.<!--QuoteEnd--></div><!--QuoteEEnd--> Not as much as makes fun of US, just hates. I've heard many fat jokes concerning overweight Americans which I immediately followed up with "Just because you haven't heard of muscle mass, doesn't mean it doesn't exist. Its not a myth, go to a gym"
<!--quoteo(post=1679938:date=Jun 1 2008, 11:28 AM:name=Geminosity)--><div class='quotetop'>QUOTE(Geminosity @ Jun 1 2008, 11:28 AM) <a href="index.php?act=findpost&pid=1679938"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->I don't think the super huge lady who almost ran me down with her lil cart while I was in Universal Studios was carrying much muscle =3<!--QuoteEnd--></div><!--QuoteEEnd--> Heh, Theme Parks are known to attract Ham Beasts. So are fast food restaurants.
I bet she WAS carrying much muscle. Think about it. I probably can't heft something that weighs as much as I do, but many fat people weigh twice as much as I. Ergo they must be stronger than me. Of course, I don't carry much muscle either.
<!--quoteo(post=1679963:date=Jun 1 2008, 02:19 PM:name=lolfighter)--><div class='quotetop'>QUOTE(lolfighter @ Jun 1 2008, 02:19 PM) <a href="index.php?act=findpost&pid=1679963"><{POST_SNAPBACK}></a></div><div class='quotemain'><!--quotec-->I bet she WAS carrying much muscle. Think about it. I probably can't heft something that weighs as much as I do, but many fat people weigh twice as much as I. Ergo they must be stronger than me. Of course, I don't carry much muscle either.<!--QuoteEnd--></div><!--QuoteEEnd--> Aw, but now-a-days they don't even need to be able to lift themselves. They can get one of those fat-scooters to haul their lard around for them.
Comments
Align: In practice is actually does still matter. Consider the 3 possibilities.
You chose a nothing box, He uncovers a nothing box. You change You win
You chose the other nothing box, He uncovers a nothing box. You change You win
You chose the correct box, he uncovers a nothings box, You change You lose.
If you change you have a 2/3 in chance.
The choice is between two boxes, each with an equal chance of containing the keys. That's similar to chosing heads or tails. The coin has an equal chance of landing on either side, so no matter which one you pick, your chances of winning are 50%.
Whether you look at it mathematically or practically, the choice is the same: There is no choice. You're at the whim of fate, and can't do anything to increase the odds. DiscoZombie got it right the first time.
EDIT: This is wrong. See my next post.
Align: In practice is actually does still matter. Consider the 3 possibilities.
You chose a nothing box, He uncovers a nothing box. You change You win
You chose the other nothing box, He uncovers a nothing box. You change You win
You chose the correct box, he uncovers a nothings box, You change You lose.
If you change you have a 2/3 in chance.<!--QuoteEnd--></div><!--QuoteEEnd-->
This would only be valid if you had any idea of more than one box.
Practically-No-Go as we say in quizspeak.
So we now have two possible scenarios: You initially choose the key box, or you initially choose an empty box. Since there are two empty boxes and one key box, you are twice as likely to choose an empty box as you are to choose the key box (meaning you have a 66% chance of choosing an empty box). Since round two will always consist of one empty box and the key box, and you have a 66% likelihood of initially having chosen an empty box, there is now a 66% likelihood that the OTHER box will be the one holding the keys. Therefore, in order to maximise your chances at winning, you should switch.
Practically-No-Go as we say in quizspeak.<!--QuoteEnd--></div><!--QuoteEEnd-->
Can you explain your reasoning behind this?
This isn't a complicated problem, it's high-school statistics (Which just so happens to be where I heard it first). Write out all the possibilities and the answer is clear...
the second rule of NS off-topic...
Well, some parts did, such as customization (one of my favorites *sob*)
Off-Topic is alive and kicking though <img src="style_emoticons/<#EMO_DIR#>/biggrin-fix.gif" style="vertical-align:middle" emoid=":D" border="0" alt="biggrin-fix.gif" /><!--QuoteEnd--></div><!--QuoteEEnd-->
Those used to be *my* forums
Well, some parts did, such as customization (one of my favorites *sob*)
Off-Topic is alive and kicking though <img src="style_emoticons/<#EMO_DIR#>/biggrin-fix.gif" style="vertical-align:middle" emoid=":D" border="0" alt="biggrin-fix.gif" /><!--QuoteEnd--></div><!--QuoteEEnd-->
The NS2 general discussion and I&S are also quite alive. I spend most of my time on the board these days trying to get people to think through on their suggestions; supporting great new concepts or changes to existing designs while pointing out the flaws in awful ideas.
If you really have problems getting it, turn it into an n problem.
You got n boxes.
So the chance that the box you choose contains the keys are 1/n.
The chance that the key is in any of the unchoosen boxes is (n-1)/n
Quizmaster opens m boxes. Which gives us: (n-1)/(n*(n-m-1)) for the chance of one of the unchoosen, unrevealed boxes containing the key.
Please remember, that m:[1;n-2]
Now that we know the mathematics behind this (which I just rethought of. I did not write them off somewhere (though I did this before in grade 13))
All we need to compare is the chance for th initial box and the chance for the potential trade box.
Assumption:
(n-1)/(n*(n-m-1)))>1/n
(n-1)/(nn-nm-n)>1/n
(nn-n)/(nn-nm-n)>1
Thats of course always the case, because n and m are >0 by definition.
But just in case that you dont see it:
nn-n=x
->
x/(x-nm)>1
Should be clear now.
As you can see it is always worth changing.
You can advance the problem, by allowing more than 1 box to be choosen, but then the result is not surprising, cause depending on how many boxes are opened, you either be better off with changing or staying.
Align: In practice is actually does still matter. Consider the 3 possibilities.
You chose a nothing box, He uncovers a nothing box. You change You win
You chose the other nothing box, He uncovers a nothing box. You change You win
You chose the correct box, he uncovers a nothings box, You change You lose.
If you change you have a 2/3 in chance.<!--QuoteEnd--></div><!--QuoteEEnd-->That's the mathematical way, yes. Not the "common sense" way, which tells us that we have no idea which box contains the key so it doesn't matter how or when or which we pick.
....<!--QuoteEnd--></div><!--QuoteEEnd-->
That was way too complicated.
I liked the simpler explanations that were suggested.
"Common sense" in this case is wrong though. You don't know which box contains the key, but that doesn't mean that it doesn't matter which one you pick, as the keys are not equally likely to be in either box. Common sense is to pick the box with higher odds, namely the one you DIDN'T pick initially.
What you describe may be common thinking for those who don't statistically analyse the problem, but their reasoning is flawed: They argue that they cannot make an informed decision because they lack information, but fail to realise that they CAN make an <u>educated guess</u> based on the information that IS available.
It might not be the "common sense way" but that does not remove the practical nature of the solution. It increases your chances of getting what you want so you should do it. Very practical.
NS2, I'm sure we'll all be back moaning/commenting (delete to taste) on the game. <img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
- Shockwave
NS2, I'm sure we'll all be back moaning/commenting (delete to taste) on the game. <img src="style_emoticons/<#EMO_DIR#>/smile-fix.gif" style="vertical-align:middle" emoid=":)" border="0" alt="smile-fix.gif" />
- Shockwave<!--QuoteEnd--></div><!--QuoteEEnd-->
*sends you a hug data packet*
Yeah pretty much all my attention is on helping in any way I can with making NS2 a positively kickass game. And that's with "spare time", which really just means "procrastinated time" on account of my generally slammed schedule.
Like I said before, when the community thins like this it's clear who the long-term members are that have true loyalty to the game and the community really are. The hosts of the party are the one's left when the drunks leave.
For the motherland!
of Canada.
of Canada.<!--QuoteEnd--></div><!--QuoteEEnd-->
O_o
Not as much as makes fun of US, just hates. I've heard many fat jokes concerning overweight Americans which I immediately followed up with "Just because you haven't heard of muscle mass, doesn't mean it doesn't exist. Its not a myth, go to a gym"
Heh, Theme Parks are known to attract Ham Beasts. So are fast food restaurants.
How strong are you?
I could totally lift like, a totally fat chick!
Aw, but now-a-days they don't even need to be able to lift themselves. They can get one of those fat-scooters to haul their lard around for them.