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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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Solution solution | Comment 1 of 14
In the following discussion, my reference to "last marble" refers to what would have been the last marble, were the drawing to continue until the 25th marble had been drawn. This allows for an easier calculation that the first color to be exhausted is the opposite color:

There is a 1/5 probability that the first drawn will be black.  The conditional probability in this case is 20/24 that the last marble drawn will be white, which is the same as saying that the black marbles will be exhausted before the white.

The other possibility is that the first drawn will be white, with probability 4/5. In this case the conditional probability will be 5/24 that the last marble will be black, which is that the white marbles will be exhausted before the black.

Thus the overall probability will be:

(1/5)*(20/24) + (4/5)*(5/24) = 40/120 = 1/3

The first paragraph preface has been added as it has been pointed out that the puzzle states drawing is terminated upon one of the colors being exhausted. Carrying out the drawing to the last (overall) marble doesn't affect which color runs out first, it just makes the probability easier to calculate.

Edited on August 12, 2015, 2:44 pm
  Posted by Charlie on 2015-08-11 15:56:22

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