Refinement monoid - Revision history

Deltahedron: link

2014-08-21T18:06:46Z

link

← Older revision		Revision as of 20:06, 21 August 2014
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	~~'''Banach's match problem'''~~ is ~~a classic problem in [[probability]] attributed to [[Stefan Banach]]~~. ~~Feller <ref> Feller, William, An Introduction to Probability Theory And Its Applications, Third Edition, Wiley, 1968, Chapter VI, section 8 </ref> says~~ that ~~the problem was inspired by a humorous reference to Banach~~'~~s smoking habit in~~ a ~~speech honouring him by H~~. ~~Steinhaus, but that it was not Banach who set the problem or provided an answer.~~		Hello and welcome. My title is Ling. Playing crochet is some thing that I've done for many years. Bookkeeping has been his working day job for a whilst. Alabama has usually been his house and his family members enjoys it.<br><br>My website [http://www.caribbeansocialnetwork.com/groups/easy-diy-repair-strategies-for-your-automobile/ extended car warranty]

	~~Suppose a mathematician carries two matchboxes at all times: one in~~ his ~~left pocket~~ and ~~one in~~ his ~~right. Each time he needs a match, he is equally likely to take~~ it ~~from either pocket~~. ~~Suppose he reaches into his pocket and discovers for the first time that the box picked is empty. If it is assumed that each of the matchboxes originally contained~~ <~~math~~>N<~~/math~~> ~~matches, what is the probability that there are exactly <math>k</math> matches in the other box?~~

	~~==Solution==~~

	Without loss of generality consider the case where the matchbox in his right pocket has an unlimited number of matches and let M be the number of matches removed from this one before the left one is found to be empty. When the left pocket is found to be empty, the man has chosen that pocket (N+1) times. Then M is the number of successes before (N+1) failures in Bernoulli trials with p=1/2, which has the [~~[negative binomial distribution]] and thus~~

	:~~<math>P[M=m] = \binom{N+m}{m}\left(\frac{1}{2}\right)^{N+1+m}<~~/~~math>~~.

	~~Returning to the original problem, we see that the probability that the left pocket is found to be empty first is <math>P[M<N+1]<~~/~~math> which equals 1~~/~~2 because both are equally likely. We see that the number K of matches remaining in the other pocket is~~

	~~:<math>P[K=k] = P[M=N~~-~~k\|M<N+1] = 2P[M=N~~-~~k] = \binom{2N~~-~~k}{N}\left(\frac{1}{2}\right)^{2N~~-~~k}</math>.~~

	~~==References==~~
	~~{{Reflist}}~~
	~~==External links==~~
	* [http://www-~~stat.stanford.edu/~susan/surprise~~/~~Banach.html Java applet]~~

	~~[[Category:Applied probability]]~~
	~~[[Category:Named probability problems]~~]

en>ChrisGualtieri: General Fixes using AWB

2013-10-24T00:54:57Z

General Fixes using AWB

← Older revision		Revision as of 02:54, 24 October 2013
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	~~The name of the author~~ is ~~Nestor~~. ~~Years ago we moved~~ to ~~Arizona~~ but ~~my spouse wants us~~ to ~~transfer~~. ~~Bookkeeping~~ is ~~what he does~~. ~~Her buddies say~~ it~~'s not great for her but~~ what ~~she enjoys performing~~ is ~~flower arranging~~ and ~~she~~ is ~~trying~~ to ~~make it a occupation~~.<br><br>~~my blog :~~: [http://~~owenphelps~~.~~com~~/~~UserProfile~~/~~tabid~~/~~355/userId/172983/Default~~.~~aspx car warranty~~]		'''Banach's match problem''' is a classic problem in [[probability]] attributed to [[Stefan Banach]]. Feller <ref> Feller, William, An Introduction to Probability Theory And Its Applications, Third Edition, Wiley, 1968, Chapter VI, section 8 </ref> says that the problem was inspired by a humorous reference to Banach's smoking habit in a speech honouring him by H. Steinhaus, but that it was not Banach who set the problem or provided an answer.

			Suppose a mathematician carries two matchboxes at all times: one in his left pocket and one in his right. Each time he needs a match, he is equally likely to take it from either pocket. Suppose he reaches into his pocket and discovers for the first time that the box picked is empty. If it is assumed that each of the matchboxes originally contained <math>N</math> matches, what is the probability that there are exactly <math>k</math> matches in the other box?

			==Solution==

			Without loss of generality consider the case where the matchbox in his right pocket has an unlimited number of matches and let M be the number of matches removed from this one before the left one is found to be empty. When the left pocket is found to be empty, the man has chosen that pocket (N+1) times. Then M is the number of successes before (N+1) failures in Bernoulli trials with p=1/2, which has the [[negative binomial distribution]] and thus

			:<math>P[M=m] = \binom{N+m}{m}\left(\frac{1}{2}\right)^{N+1+m}</math>.

			Returning to the original problem, we see that the probability that the left pocket is found to be empty first is <math>P[M<N+1]</math> which equals 1/2 because both are equally likely. We see that the number K of matches remaining in the other pocket is

			:<math>P[K=k] = P[M=N-k\|M<N+1] = 2P[M=N-k] = \binom{2N-k}{N}\left(\frac{1}{2}\right)^{2N-k}</math>.

			==References==
			{{Reflist}}
			==External links==
			* [http://www-stat.stanford.edu/~susan/surprise/Banach.html Java applet]

			[[Category:Applied probability]]
			[[Category:Named probability problems]]

en>ChrisGualtieri: /* References */Typo fixing, typos fixed: , → , using AWB

2012-07-24T03:36:28Z

References: Typo fixing, typos fixed: , → , using AWB

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The name of the author is Nestor. Years ago we moved to Arizona but my spouse wants us to transfer. Bookkeeping is what he does. Her buddies say it's not great for her but what she enjoys performing is flower arranging and she is trying to make it a occupation.<br><br>my blog :: [http://owenphelps.com/UserProfile/tabid/355/userId/172983/Default.aspx car warranty]