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Here's a problem I came across:

In a bag, there are $30$ one-color marbles of three different colors, say black, blue, and white. If we randomly take $25$ marbles out of the bag, among our picks will always be at least three white, at least five blue, and at least seven black marbles. How many marbles of each color are there in the bag originally?

From my interpretation of the wording, every sample of $25$ will have at least $3$ white, $5$ blue, and at least $7$ black. What's the most concise way to answer the question with my interpretation of the situation?

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There are $30$ total marbles. You pick out $25$, leaving $5$ in the bag. Now consider the worst case scenarios:

  • If the $5$ marbles left in the bag are all white, then you need that initially there were $8$ white marbles in order to guarantee that at least you will have $3$ marbles in your draw that are white.
  • If the $5$ marbles left in the bag are all blue, then initially you would need $10$ blue marbles in total to guarantee at least $5$ blue in your draw.
  • If the $5$ marbles left in the bag are all black, then you need that initially there were $12$ white marbles to guarantee at least $7$ black in the draw.

Adding up: $8$ (white) + $10$ (blue) +$12$ (black) =$30$ marbles.

This will guarantee you the result.

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Even if all $5$ unsampled marbles are white, we still have $3$ white marbles in the sample, so there must be $3+5=8$ white marbles. Similarly, there are $10$ blue and $12$ black. Note that the colors changed in the middle of the problem; I'm using the second set.

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