Riddle Me This, Batman!

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AuthorTopic: Riddle Me This, Batman!
Law Bringer
Member # 335
Profile Homepage #75
In terms of ratios, it tends to 1:1. The more families you have the closer it will be to 1:1.

In terms of majority, it turns out we don't agree. For any given single family the probabilities break down like this:

If families are capped at one child, there's an even chance of a boy or a girl and that's all. Both majorities are equally likely.

With any higher cap, there is a 50% chance of having one boy, which means the boy is the majority in the family. There is a 25% chance of having one girl and one boy, so neither is the majority. The remaining 25% is two or more girls, which leaves girls in the majority.

That's true for one family, which means it's true for any number of families. The more families you have the closer the observed ratio will be to 2 male majorities per female majority, but on average it will be 2:1. It will not come out even no matter how many or how few families you have.

—Alorael, who is glad that at least the particular details of the disagreement have been worked out.
Posts: 14579 | Registered: Saturday, December 1 2001 08:00
Agent
Member # 2820
Profile #76
To be continued...

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Thuryl: I mean, most of us don't go around consuming our own bodily fluids, no matter how delicious they are.
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Alorael: War and violence would end if we all had each other's babies!
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Drakefyre: Those are hideous mangos.
Posts: 1415 | Registered: Thursday, March 27 2003 08:00
Board Administrator
Member # 1
Profile Homepage #77
/agree Aloreal.

Oh, and stop using the word "infinite". Slap your hand when you type it. It will only confuse everyone. The math for probability distributions on infinite serieses is complicated and skunky and involves a high-end thing called a "measure" and is best avoided when unnecessary.

Use the words "arbitrarily large" instead. This means big as you like, but still a finite quantity you can do sensible math on. (And people who remember your calculus, yes, we are basically talking about limits as n -> infinity here.)

- Jeff Vogel

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Official Board Admin
spidweb@spiderwebsoftware.com
Posts: 960 | Registered: Tuesday, September 18 2001 07:00
Agent
Member # 2820
Profile #78
As a note, the series math for this problem could be done by anyone who understands induction.

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Thuryl: I mean, most of us don't go around consuming our own bodily fluids, no matter how delicious they are.
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Alorael: War and violence would end if we all had each other's babies!
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Drakefyre: Those are hideous mangos.
Posts: 1415 | Registered: Thursday, March 27 2003 08:00
Law Bringer
Member # 335
Profile Homepage #79
Since it's really not much fun to try to write out math on forums (or with most word processors) I think hand-waving infinite to mean the limit as n approaches infinity works fine. Since we're playing the math close to our chests, apparently, it's all fuzzy anyway.

quote:
Originally written by Spidweb:

/agree Aloreal.
—Alorael, who has decided to exercise discretion in order to be valorous in this particular instance.
Posts: 14579 | Registered: Saturday, December 1 2001 08:00
Agent
Member # 2820
Profile #80
Did ya get my PM?

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Thuryl: I mean, most of us don't go around consuming our own bodily fluids, no matter how delicious they are.
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Alorael: War and violence would end if we all had each other's babies!
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Drakefyre: Those are hideous mangos.
Posts: 1415 | Registered: Thursday, March 27 2003 08:00
Electric Sheep One
Member # 3431
Profile #81
Since we're all having so much fun with this one, or at any rate the number of posts in this thread is becoming arbitrarily large, perhaps it's time for the Prisoner's Dilemma.

Two partners in crime are being tried separately. A sharp prosecutor makes the same proposal to each one: rat on the other, for a reduced sentence of 5 years, so that the prosecutor can score a major conviction on the other one and get him put away for 20. If neither crook confesses anything, circumstantial evidence will send them both to jail for 10 years. If they both try to rat on each other, the credibility loss will make the biggest case fall apart, but they'll both look bad enough to pull 15 years.

So each prisoner has the choice to collaborate with the prosecution or not. If both collaborate they get 15 each; if both keep quiet, they get 10; if only one collaborates, he gets 5 and the other gets 20.

The orthodox, game theory answer to this problem is clear. Prisoner A should figure that if prisoner B collaborates, he should collaborate too to get 15 instead of 20 years. And if prisoner B doesn't, he should collaborate to get 5 instead of 10. So either way, he should collaborate.

The funny part is that both prisoners' reasoning should be the same, so both should collaborate, and both get 15 years. Whereas if they had not bothered with game theory, and just stuck to the crooks' code, they would both have kept quiet and gotten away with 10 years each. Somehow game theory makes the outcome worse.

If you haven't heard this one before, it's actually important. It is used as a simple model for all kinds of dismayingly bad real world problems.

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We're not doing cool. We're doing pretty.
Posts: 3335 | Registered: Thursday, September 4 2003 07:00
Agent
Member # 2820
Profile #82
I remember discussing this in AP Psychology. However, my teacher was a bit flaky on the subject and no one truly understood the situation. If I remember correctly, it is aptly called a dilemma because it all depends on the reasoning of the other person.

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Thuryl: I mean, most of us don't go around consuming our own bodily fluids, no matter how delicious they are.
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Alorael: War and violence would end if we all had each other's babies!
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Drakefyre: Those are hideous mangos.
Posts: 1415 | Registered: Thursday, March 27 2003 08:00
Law Bringer
Member # 6785
Profile #83
I liked the time Dilbert in the animated TV series was faced with this dilemma and decided not to talk only to find out that his two engineering friends had ratted him out to escape punishment.
Posts: 4643 | Registered: Friday, February 10 2006 08:00
E Equals MC What!!!!
Member # 5491
Profile Homepage #84
1 - 0.999... = ?

If you know this one already, keep quiet.

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SupaNik: Aran, you're not big enough to threaten Ash. Dammit, even JV had to think twice.
Posts: 1861 | Registered: Friday, February 11 2005 08:00
...b10010b...
Member # 869
Profile Homepage #85
quote:
Originally written by Ash Lael:

1 - 0.999... = ?

If you know this one already, keep quiet.

Oh no. No. Please no. This one always, always gets completely asinine.

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The Empire Always Loses: This Time For Sure!
Posts: 9973 | Registered: Saturday, March 30 2002 08:00
Electric Sheep One
Member # 3431
Profile #86
The obvious answer is: ...

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We're not doing cool. We're doing pretty.
Posts: 3335 | Registered: Thursday, September 4 2003 07:00
Agent
Member # 2820
Profile #87
I love this one. I find it so commonly known sometimes, though, that it is no longer a puzzle when you are with a group of people.

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Thuryl: I mean, most of us don't go around consuming our own bodily fluids, no matter how delicious they are.
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Alorael: War and violence would end if we all had each other's babies!
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Drakefyre: Those are hideous mangos.
Posts: 1415 | Registered: Thursday, March 27 2003 08:00
Guardian
Member # 6670
Profile Homepage #88
By Ash:
quote:
1 - 0.999... = ?
I find it funny when people try to explain this one. This question's become a sort of mathematical psychoanalysis for me. Either somebody will spend two minutes playing around with summation notation to prove their point, or they'll spend two seconds drawing an analogy.

Don't worry, I won't give the answer. It's fun to watch people squirm with n -> inf.

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IF I EVER BECOME AN EVIL OVERLORD:
I will not outsource core functions.
Posts: 1509 | Registered: Tuesday, January 10 2006 08:00
Triad Mage
Member # 7
Profile Homepage #89
I use fractions.

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"At times discretion should be thrown aside, and with the foolish we should play the fool." - Menander
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Drakefyre's Demesne - Happy Happy Joy Joy
Encyclopedia Ermariana - Trapped in the Closet
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You can take my Mac when you pry my cold, dead fingers off the mouse!
Posts: 9436 | Registered: Wednesday, September 19 2001 07:00
Guardian
Member # 6670
Profile Homepage #90
By Drakey:
quote:
I use fractions.
That would be the two second method. But now that you've said that, we won't have anybody wasting their time putting up a half-page explaination with sigma notation! Curse you!

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IF I EVER BECOME AN EVIL OVERLORD:
If I ever build a device to transfer the hero's energy into me, I will make sure it cannot operate in reverse.
Posts: 1509 | Registered: Tuesday, January 10 2006 08:00
Law Bringer
Member # 335
Profile Homepage #91
In other news, I think I've figured out the odds for total majority:

There is a 50% chance of a male majority, a 3^n/4^n chance of a tie, and (obviously) a 1/2-3^n/4^n chance of female majority counting all the children of n families.

[Edit: Retraction removed and expression fixed.]

—Alorael, who knows that you end up with 0.0...01 after doing the arithmetic. It's a pity that the universe can't handle that many bits and rounds it to zero.

[ Sunday, August 20, 2006 19:18: Message edited by: Eyes Only ]
Posts: 14579 | Registered: Saturday, December 1 2001 08:00
Agent
Member # 2820
Profile #92
10^-n approaches 0 as n -> infinity, so it all makes sense. The more boggling variation is to ask what 0.8888... + 0.1111... equals.

EDIT: By the way, relief on seeing my boy/girl argument understood at last. I had all but left it to the vultures already. The ratio for the tie seems a little off to me, but the conclusion is nonetheless the same.

[ Sunday, August 20, 2006 19:49: Message edited by: Garrison ]

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Thuryl: I mean, most of us don't go around consuming our own bodily fluids, no matter how delicious they are.
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Alorael: War and violence would end if we all had each other's babies!
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Drakefyre: Those are hideous mangos.
Posts: 1415 | Registered: Thursday, March 27 2003 08:00
Infiltrator
Member # 2836
Profile #93
quote:
Originally written by Garrison:

The more boggling variation is to ask what 0.8888... + 0.1111... equals.
It equals 0.99999 etc which can be easily proved to equal 1.
Posts: 587 | Registered: Tuesday, April 1 2003 08:00
Electric Sheep One
Member # 3431
Profile #94
Isn't 0.999... more specifically equal to the limit as you approach 1 from below? It seems as though it ought to be. So then 1/(1-0.999...) would be positive infinity, with unambiguous sign.

So how come we can't do things like this to make the limit as you approach 1 from above? Writing 1.000 ... 001 doesn't really work, does it?

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We're not doing cool. We're doing pretty.
Posts: 3335 | Registered: Thursday, September 4 2003 07:00
Law Bringer
Member # 335
Profile Homepage #95
quote:
Originally written by Garrison:

The ratio for the tie seems a little off to me, but the conclusion is nonetheless the same.
Those nice cases of 0 (or not), 1, and the infinite limit are the important ones anyway. The rest is just messy real-world details.

—Alorael, who wonders if rounded insignificant digits go to heaven. Or do they just end up in a hellish nightmare or puncuation, letters, and other oddments with nary a 3 in sight?
Posts: 14579 | Registered: Saturday, December 1 2001 08:00
Post Navel Trauma ^_^
Member # 67
Profile Homepage #96
quote:
Originally written by Student of Trinity:

Isn't 0.999... more specifically equal to the limit as you approach 1 from below? It seems as though it ought to be. So then 1/(1-0.999...) would be positive infinity, with unambiguous sign.

So how come we can't do things like this to make the limit as you approach 1 from above? Writing 1.000 ... 001 doesn't really work, does it?

0.999... isn't specifically more equal to anything other than plain old 1. It's not "approaching" anything, it's already there. It's just that decimal notation has the fun little property that some numbers have more than one representation.

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Barcoorah: I even did it to a big dorset ram.

desperance.net - Don't follow this link
Posts: 1798 | Registered: Thursday, October 4 2001 07:00
Electric Sheep One
Member # 3431
Profile #97
I did not say it was approaching; I said it was the limit. I would be surprised if you could define infinite decimal representations without some invocation of limits. If you can, go ahead; I'd be happy to see it.

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We're not doing cool. We're doing pretty.
Posts: 3335 | Registered: Thursday, September 4 2003 07:00
La Canaliste
Member # 5563
Profile #98
Khothie might be more inclined to perform that feat with an invocation of Cthulhu and the nameless ones.

He is *quite* good at maths though.

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I am a mater of time and how .

Deep down, you know you should have voted for Alcritas!
Posts: 387 | Registered: Tuesday, March 1 2005 08:00
Post Navel Trauma ^_^
Member # 67
Profile Homepage #99
All I really meant was that there is nothing "from below" about 0.999..., so you can't say "1/(1-0.999...) = +ve infinity"

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Barcoorah: I even did it to a big dorset ram.

desperance.net - Don't follow this link
Posts: 1798 | Registered: Thursday, October 4 2001 07:00

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