Two Plus Two Older Archives  

Go Back   Two Plus Two Older Archives > General Gambling > Probability

Reply
 
Thread Tools Display Modes
  #1  
Old 10-20-2002, 09:09 AM
irchans irchans is offline
Senior Member
 
Join Date: Sep 2002
Posts: 157
Default Sock Drawer

A drawer contains red socks and black socks. When two socks are drawn at random, the probabilitiy that both are red is 1/2. (a) How small can the number of socks in the drawer be? (b) How small if the black socks is even? (c) How small if there are at least 30 socks?


Please put answer-a) or answer-b) or answer-c) in the subject if you post an answer.




(Mosteller)
Reply With Quote
  #2  
Old 10-20-2002, 02:00 PM
Guest
 
Posts: n/a
Default Re: Sock Drawer

I don't quite understand your question as worded but that has neve stopped me from answering yet! [img]/forums/images/icons/smile.gif[/img]

A) 3
B) 4
C) 2

Jimbo

Now if you are asking what is the minimum number of socks you must draw before you are assured of retrieving 2 red socks my answer to C) would be 16. If you are asking how many before it is more likely than not that you will have 2 red socks, I'll leave that to the real mathmaticians.
Reply With Quote
  #3  
Old 10-20-2002, 07:51 PM
PseudoPserious PseudoPserious is offline
Senior Member
 
Join Date: Oct 2002
Posts: 151
Default Re: Sock Drawer - ans (a,b,c)

Nifty.

a) 4 (r=3, b=1)
b) 21 (r=15, b=6)
c) 120 (r=85, b=35)

Let (r/(r+b))((r-1)/(r+b-1)) = .5. Simplify. Find the roots that satisfy the given constraints.

PP
Reply With Quote
  #4  
Old 10-20-2002, 08:30 PM
BruceZ BruceZ is offline
Senior Member
 
Join Date: Sep 2002
Posts: 1,636
Default Re: Sock Drawer - ans (a,b,c)

This reduces to finding integer solutions to t(t-1) = 2r(r-1). I also found the above solutions using excel. I'm guessing we can use number theory to find all the solutions in general, right irchans?
Reply With Quote
  #5  
Old 10-21-2002, 05:32 AM
irchans irchans is offline
Senior Member
 
Join Date: Sep 2002
Posts: 157
Default Re: Sock Drawer - ans (a,b,c)

BruceZ, Pseudo,

You got the right answers! Number Theory is certainly involved in the "general answer" which is beyond me. Mosteller writes, "... we would be wise to appreciate that this is a problem in the theory of numbers. It happens to lead to a famous result in Diophantine Analysis obtained from Pell's equation."

Pell's equation is

n x^2 + 1 = y^2 .

I don't see how this is related to the sock problem. Mollester gives a reference (Elementary theory of numbers by LeVeque.) If I get some time tomorrow, I will look it up.

Cheers, Irchans
Reply With Quote
  #6  
Old 10-22-2002, 01:09 AM
Bozeman Bozeman is offline
Senior Member
 
Join Date: Sep 2002
Location: On the road again
Posts: 1,213
Default Re: Sock Drawer - ans (a,b,c)

Mollester gives a reference

Is this a Freudian slip?
Reply With Quote
  #7  
Old 09-16-2003, 03:47 PM
thylacine thylacine is offline
Senior Member
 
Join Date: Jul 2003
Posts: 294
Default Re: Sock Drawer - ans (a,b,c)

I stumbled across this thread via the more recent thread "Probability Problem (non-poker)."

BruceZ says we are solving t(t-1) = 2r(r-1).

irchans says he reads of a connection to Pell's equation: n x^2 + 1 = y^2 .

The connection is this. Let T=2t-1, R=2r-1.

Then t(t-1) = 2r(r-1) becomes 2R^2=T^2+1.

(I don't know how to solve these or what is known.)


Reply With Quote
  #8  
Old 09-16-2003, 04:25 PM
thylacine thylacine is offline
Senior Member
 
Join Date: Jul 2003
Posts: 294
Default all(?) solutions

It just struck me that solving 2R^2=T^2+1 is related to the continued fraction for sqrt(2).

The bottom line is that if (R,T) is a solution, then so is (3R+2T,4R+3T).

Starting with (R,T)=(1,1) you generate infinitely many solutions, and I think these might be all.

So Solutions are (R,T)=(1,1), (5,7), (29,41), (169,239), ...

Then translate back to get solutions (r,t) to t(t-1) = 2r(r-1).

Reply With Quote
Reply

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off

Forum Jump


All times are GMT -4. The time now is 10:08 PM.


Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2021, vBulletin Solutions Inc.