Banach's matchbox problem: Difference between revisions
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==Solution== |
==Solution== |
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Let <math>E</math> denote the event that the man discovers the matchbox in his right pocket is empty and there are <math>k</math> matches in the matchbox in his left pocket. This event occurs only if the <math>(N + 1)</math>th choice of the matchbox in his right pocket is made at the <math>N + 1 + N - k</math> trial. |
Let <math>E</math> denote the event that the man discovers that the matchbox in his right pocket is empty and there are <math>k</math> matches in the matchbox in his left pocket. This event occurs only if the <math>(N + 1)</math>th choice of the matchbox in his right pocket is made at the <math>N + 1 + N - k</math> trial. :) |
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Hence <math>E</math> is a [[random variable]] with the [[negative binomial distribution]], with parameters |
Hence <math>E</math> is a [[random variable]] with the [[negative binomial distribution]], with parameters |
Revision as of 14:49, 5 February 2010
Banach's match problem is a classic problem in probability attributed to Stefan Banach.
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 that the box picked is empty. If it is assumed that each of the matchboxes originally contained matches, what is the probability that there are exactly matches in the other box?
Solution
Let denote the event that the man discovers that the matchbox in his right pocket is empty and there are matches in the matchbox in his left pocket. This event occurs only if the th choice of the matchbox in his right pocket is made at the trial. :)
Hence is a random variable with the negative binomial distribution, with parameters don't joke
and so
Since it is equally likely that the matchbox found to be empty is in the left pocket, the desired probability is
im not even joking, this is the right answer
References
- Ross, Sheldon (2006). A First Course in Probability (Seventh edition ed.). pp. pp. 176—177. ISBN 0131856626.
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