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Question

I have a deck of $n$ cards where $m$ of them are marked. I ask you to pick out $x$ cards from the deck (without replacement). What's the probability that you pick at least $k$ of the marked cards?

Obviously, $x \le n$ and $k \le m$.

What I've done so far:

In my head, I'm thinking of the problem as a balls and urns problem, where one urn contains the "marked balls" and the other urn contains the "unmarked balls". Since you're choosing a ball randomly, you'll have a particular chance of picking from one of the two urns each time you pick, based on how many balls are in each urn.

However, since we're picking without replacement the probability that you pick from a particular urn changes each time you take out a ball. And that complexity signals to me that I'm thinking about this problem wrong.

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  • $\begingroup$ I'm too busy to write up an answer, but this is a well-known distribution called the hypergeometric distribution $\endgroup$
    – Pro Q
    Commented Mar 21, 2021 at 9:21

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As you also commented, this is related to hypergeometric distribution. Probability of picking exactly $k$ marked balls is:

$$p_k=\frac{{m\choose k}{n-m\choose x-k}}{n\choose x}$$

And, probability of at least $k$ marked balls is $\sum_{l=k}^m p_l$.

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