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Simpler probability problem? (Posted on 2015-08-11) Difficulty: 1 of 5
An urn contains 5 black and 20 white marbles. They are to be drawn one at a time until all of one color has been exhausted.

What is the probability that the color of the first draw correctly predicts the first color exhausted?

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re(2): I think I'm beginning to understand what Ady's saying Comment 14 of 14 |
(In reply to re: I think I'm beginning to understand what Ady's saying by Ady TZIDON)

If there are b black marbles and w white marbles, the probability that the first color drawn will be the color that gets exhausted first is:

b/(b+w) * w/(b+w-1) + w/(b+w) * b/(b+w-1)

You've noted that these two terms are equal in the problem at hand. But it now is clear (I didn't think about it before) that these would always be equal: Each term is b*w/((b+w)*(b+w-1)) and therefore the overall probability that the first to be drawn matches the first to be exhausted is just twice this: 2*b*w/((b+w)*(b+w-1)).

 b     w      prob of match
 5    20      1/3
25     1      1/13
17     7      119/276
13    13      13/25

A couple of these are of special interest:

When there's only one white marble, then of course if the white marble is drawn first it is certain to be the first to be exhausted because it is already exhausted by that one draw.

Only in the case of equality, 13,13 is the probability of a match greater than 1/2, which it of course is, as that first draw puts it in the minority and therefore in greater danger of extinction.

One thing about this reminds me of Simpson's paradox. Although not the same thing, it is similar to that paradox. In the case of the original puzzle, although a first draw of a black marble makes it more likely that  black marbles would be the first to be exhausted and a first draw of white makes it more likely that white would be the first exhausted (than a priori), it is 2/3 probable that the first drawn does not match the first to be exhausted. This of course is attributable to the high likelihood that white is the first chosen and that black is the first to be exhausted.

  Posted by Charlie on 2015-08-13 09:20:42
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