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Matching Matchbox Muse (Posted on 2010-03-30) Difficulty: 3 of 5
Professor X smokes a pipe. He carries two identical matchboxes, originally containing 20 matches each. When he lights his pipe, he chooses a matchbox at random and lights his pipe with one match and discards the used match.

There will eventually arise an occasion when he first selects a matchbox with only one match in it. At this point, what is the expected number of matches in the other box?

No Solution Yet Submitted by K Sengupta    
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Solution Corrected Solution after Charlie's comments | Comment 9 of 15 |

As Charlie justly pointed out - the summations regarding the number of remaining matches N , should be taken over N=1 to 20, instead of N=2 to 20. This would cover the case of 1 match remaining in a box, but the professor choosing the other box in which also one match remained.

so the corrected solution is the following :

The event which stops the process is the occurrence of a single match in either box, while in the other one remain N matches ( N being between 1 and 20). The number of drawn matches, of the 40 existing ones, is therefore:    40-(N+1)=(39-N).

Denoting:  

AN=The number of possible different sequences of drawn matches from the 2 boxes, given the number of drawn matches from box A or box B, is 19, and the total group of drawn matches is (39-N).

The probability of the occurrence of a specific N will therefore be :

(1)   Pan = AN / (summation over N=1 to N=20 of AN)

AN will equal the number of combinations of choosing 19 entities out of a group of (39-N), which is :

(2)  AN=2*(39-N)!/(20-N)!/19!

The factor 2  takes care of the possibility that the box with one match may be either box A or box B.

Substituting eq.(2) into eq.(1) gives Pan as function of N, and allows the computation of the requested expectance E as :<o:p></o:p>

E=summation over N=1 to N=20 of(Pan*N)

A simple Matlab program computing this, gives the result :

 

                                    E = 1.9048

 

  Charlie - as you can see this corrected result is even further away from your solution, the reason having been explained in my first comment to you


 


  Posted by Dan Rosen on 2010-06-29 15:00:37
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