What is the probability of this?

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....

S_David

OK, I will try to figure it out. Thanks

viraltux

No problem, but anyway, the answer for your new problem, assuming equally likely every cardinality, is [itex]K^{-M}[/itex].
I think you've just suffered enough

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:
[itex] {{K}\choose{k}}^{-M+1}[/itex]

viraltux

The original problem was:
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:

[tex]\frac{1}{K+1}\sum_{k=0}^K {{K}\choose{k}}^{-M}[/tex]

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
[itex] {{K}\choose{k}}^{-M}[/itex] = 2^-2 = 1/4
I say [itex] {{K}\choose{k}}^{-M+1}[/itex] = 2^-1 = 1/2
Think about it.

viraltux

Oh, by 'prespecified' k you meant the actual set contained within the k balls, yes, then you're right.

ClifDavis

David, I understand that this is not the actual problem you want to solve. What is the problem you want to solve?

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?

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???

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

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.

S_David

All balls are equiprobable to be picked up. Is it still not complete?

haruspex

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.

haruspex

No, it's still not complete. That piece of information all sets of k balls are equally likely once k is chosen. You still haven't said how k is chosen.

S_David

You can assume a distribution to simplify it. I just need to see the idea.

haruspex

In that case I select a binomial distribution. This means I can rephrase it as "each ball is chosen independently with probability p".
The probability that a given ball is selected by all M is then p^M.
The number of balls selected by all is binomial with parameter p^M.
However, I have violated the statement that k must be 1 to K. I've made it 0 to K. Is that alright?