Again with the Marbles

[m_the0ry]

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First, imagine there's a pile of 100 marbles. Some of these marbles are white, and some are black (order doesn't matter). It's unknown how many marbles there are of each color. Now, n marbles are taken at random from the pile. All n marbles are white.

The question: What is the probability k that all 100 of the marbles are white for any given n?


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As n increases, so does the probability that all are white. Equal in probability is the scenario that all remaining marbles are black.

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Does it have to do with the sum of all the probabilities of drawing n white marbles with different amounts of black marbles? My guesses here are, n=90 k=.89, n=50 k=.02, n=1 k=(way low)

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The only notable correlation n has with k is that k = 100-n - the only way the variables are linked is in the sample size of k. That is to say that the problem can be equivalently worded as:

"I have k marbles, what is the probability that all are white as k varies from 1-100."

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Second problem. Same situation. Again, n marbles are taken from the pile. However, in this case, the white marbles are taken first. So if there are 4 white and 96 black marbles in the pile, with n=5 the 4 white marbles will be drawn first, then a black marble. Whites are always first!

Now the question: if all n marbles are white, what's the probability now that all the rest of the marbles are white? My intuition says in this case it's a strict 2^(n-100) (that no new information about the "hidden" marbles is available and they can be either black or white). Am I wrong?

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You are correct.