- #36

viraltux

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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?

The last time

**S_David**explained the problem it seemed he is worried about "the cardinality of the intersection" which I think hardly relates to the "The number of balls selected by all"

Rephrasing the problem this way turns it into a different one. When a problem states thing like "Each person will pull a number of balls" the most common approach is to assume every number is equally likely. It would be like asking people to choose a number from 0 to K, if we do not consider things like psychological biases (like people choosing their favorite number) the most reasonable is to assume every number is equally likely. The moment you assign a probability

**p**to a ball to be chosen then it implies things like "it is more likely people choose 4 balls than 5", which I see this nowhere in the problem. Not to mention that with this approach you need to estimate p which is not given by the problem.

Also, in any case, when a given ball is selected by all M is then p

^{M}, but the probability that all M select the same ball is p

^{M-1}... unless the problem has changed again which wouldn't be surprising at this point. :tongue:

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

I don't think there is information missing in this problem, the key difficulty lays in the "cardinality of this intersection" part. If we had the distribution for the intersection of cardinalities of M subsets within a set of cardinality K the problem is done.

I did calculate such distribution of

**m**for any given cardinalities C

_{1}and C

_{2}considering M=2 and is already huge, and for M>2 it only gets worse :tongue:

I think would I go Montecarlo on this one.