## What is the probability of this?

 Quote by S_David What if they are not equally likely? Why p does not appear in the probability? Actually, this is not the real question. The real question is: if each person pulls different number of balls, what is the probability that the cardinality of the intersection between all users is m?
It does not appear because every ball is supposed to have the same probability and that simplifies the problem.

In the new stated problem, again, every ball has the same probability! so you already know your new solution won't have any p either....

 Quote by viraltux It does not appear because every ball is supposed to have the same probability and that simplifies the problem. In the new stated problem, again, every ball has the same probability! so you already know your new solution won't have any p either....
OK, I will try to figure it out. Thanks

 Quote by S_David OK, I will try to figure it out. Thanks
No problem, but anyway, the answer for your new problem, assuming equally likely every cardinality, is $K^{-M}$.
I think you've just suffered enough

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 Quote by viraltux OK, the solution for the this problem as it is stated is ${{K}\choose{k}}^{-M}$
Not quite. That's the prob they all pick the same prespecified k. But it could be any k, so long as they all pick the same k:
${{K}\choose{k}}^{-M+1}$

 Quote by haruspex Not quite. That's the prob they all pick the same prespecified k. But it could be any k, so long as they all pick the same k: ${{K}\choose{k}}^{-M+1}$
The original problem was:
 Suppose that there are M persons, and K balls number from 1 to K. Each person pulls k balls at the same time, and return them back. What is the probability that all persons pull the same k balls?
That pretty much specifies the number of balls that are extracted: k

But even if you transform the problem and turn "each person pulls k" into "each person pulls the same number" being those numbers equally likely, and the question "What is probability that all persons pull the same k balls?" into "What is probability that all persons pull the same balls?" Then the answer would be:

$$\frac{1}{K+1}\sum_{k=0}^K {{K}\choose{k}}^{-M}$$
 Recognitions: Homework Help Science Advisor Let's take a simple example. K=2, k=1, M=2. Each takes one ball and replaces it. What's the probability they take the same ball? You say ${{K}\choose{k}}^{-M}$ = 2-2 = 1/4 I say ${{K}\choose{k}}^{-M+1}$ = 2-1 = 1/2 Think about it.

 Quote by haruspex Let's take a simple example. K=2, k=1, M=2. Each takes one ball and replaces it. What's the probability they take the same ball? You say ${{K}\choose{k}}^{-M}$ = 2-2 = 1/4 I say ${{K}\choose{k}}^{-M+1}$ = 2-1 = 1/2 Think about it.
Oh, by 'prespecified' k you meant the actual set contained within the k balls, yes, then you're right.

 Quote by S_David Hi, Suppose that there are M persons, and K balls number from 1 to K. Each person pulls k balls at the same time, and return them back. What is the probability that all persons pull the same k balls? Thanks
David, I understand that this is not the actual problem you want to solve. What is the problem you want to solve?

 Quote by ClifDavis David, I understand that this is not the actual problem you want to solve. What is the problem you want to solve?
If each person pulls different number of balls, what is the probability that the cardinality of the intersection between all users is m?

 Quote by S_David If each person pulls different number of balls, what is the probability that the cardinality of the intersection between all users is m?
Okay let me restate the problem to see if I understand it correctly.

There are M people and K distinct balls. Each of those M people will pick a number k at random between one and K inclusive and then pick k balls at random. An reliable observer will record which balls they picked. Then the selected balls are put back for the next person. After all M of these people have done this the reliable observer informs us that none of the M people picked the same number k of balls to select.

Someone gives us an integer m. We want to give them the odds that the intersection of the ball picks contains exactly m balls.

Is this a correct statement of the problem???

 Quote by ClifDavis Okay let me restate the problem to see if I understand it correctly. There are M people and K distinct balls. Each of those M people will pick a number k at random between one and K inclusive and then pick k balls at random. An reliable observer will record which balls they picked. Then the selected balls are put back for the next person. After all M of these people have done this the reliable observer informs us that none of the M people picked the same number k of balls to select. Someone gives us an integer m. We want to give them the odds that the intersection of the ball picks contains exactly m balls. Is this a correct statement of the problem???
Ok, let me elaborate more: We have M person and K distinct balls numbered from 1 to K. Each person will pull a number of balls, i.e.: the number of picked balls is not known a priori. Then he will return the balls to the next person (after taking the balls' numbers by a reliable observer) who will do the same thing. At the end we will find the intersection between the balls picked by all persons. I need to find the probability that the cardinality of this intersection is m, where m is an integer between 1 and K.

I hope it is more clear now.

Thanks

 Quote by S_David Ok, let me elaborate more: We have M person and K distinct balls numbered from 1 to K. Each person will pull a number of balls, i.e.: the number of picked balls is not known a priori. Then he will return the balls to the next person (after taking the balls' numbers by a reliable observer) who will do the same thing. At the end we will find the intersection between the balls picked by all persons. I need to find the probability that the cardinality of this intersection is m, where m is an integer between 1 and K. I hope it is more clear now. Thanks
 Quote by S_David Ok, let me elaborate more: We have M person and K distinct balls numbered from 1 to K. Each person will pull a number of balls, i.e.: the number of picked balls is not known a priori. Then he will return the balls to the next person (after taking the balls' numbers by a reliable observer) who will do the same thing. At the end we will find the intersection between the balls picked by all persons. I need to find the probability that the cardinality of this intersection is m, where m is an integer between 1 and K. I hope it is more clear now. Thanks
In order to answer the question it is not necessary to know the number of picked balls a priori. It is however necessary to know the probability distribution of the number of balls picked.

If there are M=2 people and K=5 distinct balls and there is a very high probability of 4 or 5 balls getting picked and none at all of 1 or 2 balls, then it is a very different situation than if there is no probability of 4 or 5 balls being picked and a high probability ofr 1 or 2 balls.

If I understand your question, there is not enough information to answer it.

 Quote by ClifDavis In order to answer the question it is not necessary to know the number of picked balls a priori. It is however necessary to know the probability distribution of the number of balls picked. If there are M=2 people and K=5 distinct balls and there is a very high probability of 4 or 5 balls getting picked and none at all of 1 or 2 balls, then it is a very different situation than if there is no probability of 4 or 5 balls being picked and a high probability ofr 1 or 2 balls. If I understand your question, there is not enough information to answer it.
All balls are equiprobable to be picked up. Is it still not complete?
 Recognitions: Homework Help Science Advisor ClifDavis is right, of course: you need to specify the probability distribution for k. But it sounds from an earlier post that k is supposed to have a uniform distribution from 1 to K. If so, it sounds like a very nasty problem.

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