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Probem 7(a) on page 116. A paraphrase of the problem. An urn contains a finite nonempty set B of black balls and a finite
 

Summary: Probem 7(a) on page 116.
A paraphrase of the problem. An urn contains a finite nonempty set B of black balls and a finite
nonempty set W of white balls and nothing else. Balls are drawn from the urn without replacement until
the remaining balls are of the same color. What is the probability that the remaining ball(s) are white?
Solution. Let A = B W and let M = {1, . . . , |A| - 1}. For each m M let
Em
be the set of x (A)m such that
xm B and |{i {1, . . . , m} : xi B}| = |B|;
let
Fm
be the set of x (A)m such that
xm W and |{i {1, . . . , m} : xi W}| = |W|.
and let
Sm = Em Fm.
Note that whenever m M we have Em = if m < |B| and Fm = if m < |W|. Evidently, the family
{Sm : m M} is disjointed and
S =
|A|-1
m=1 Sm
is a reasonable sample space for this experiment. For any m M, Em is the event the that the m-th ball

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics