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Topic: Molina's Urns (Read 924 times) |
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william wu
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Molina's Urns
« on: May 23rd, 2003, 4:05am » |
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Two urns contain the same total numbers of balls, some blacks and some whites in each. From each urn, n balls are drawn one at time with replacement, where n >= 3. Determine the value of n and the compositions of the urns such that the probability only white balls are drawn from the first urn is equal to the probability that the drawing from the second urn is either all white or all black.
Author: E.C. Molina.
[edit] Revised wording for added clarity. 8/14/2004 - wwu [/e]
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| « Last Edit: Aug 13th, 2004, 4:49pm by william wu » |
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ThudnBlunder
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Re: Molina's Urns
« Reply #1 on: May 23rd, 2003, 8:55am » |
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Doesn't (the truth of) Fermat's Last Theorem imply that it's impossible?
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| « Last Edit: May 23rd, 2003, 10:43pm by ThudnBlunder » |
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jonderry
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Re: Molina's Urns
« Reply #3 on: Aug 13th, 2004, 3:21pm » |
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Can't we relax the constraint that the number of balls is the same in both and the answer will still be:
it is impossible
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Sir Col
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Re: Molina's Urns
« Reply #4 on: Aug 13th, 2004, 4:32pm » |
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When you say, "with replacement", do you mean, (i) n balls are taken, distribution of colours observed, then replaced, or, (ii) one ball is taken, its colour is observed, and it is replaced, and this is repeated n times?
I am guessing it is (ii), otherwise replacement would be irrelevant in the context of this particular question...
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Let number of white balls in 1st urn be w1, and t1 balls in total.
Let number of white balls in 2nd urn be w2, black balls be b2, and t2 balls in total.
P(n white from 1st)=(w1/t1)n
P(n white from 2nd)=(w2/t2)n
P(n black from 2nd)=(b2/t2)n
Equating these, (w1/t1)n = (w2/t2)n + (b2/t2)n
Multiplying through by (t1*t2)n, (w1*t2)n = (w2*t1)n + (b2*t1)n.
Let x=w1*t2, y=w2*t1, and z=b2*t1.
As x, y, and z are integer, we are looking for an integral solution to the equation xn = y2 + zn.
By Fermat's Last Theorem we know that no solution exists, for n>=3, hence it is not possible to setup the urns in such a way.
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william wu
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Re: Molina's Urns
« Reply #5 on: Aug 13th, 2004, 4:50pm » |
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Yes, interpretation (ii) is correct. "With replacement" is vacuous in interpretation (i) since then there is only one drawing. Thanks for the remark; it was a bit unclear and I have revised the wording for more clarity. And congrats on the solution 
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