It's about distribution or arrangement of $m$ distinct or identical balls into $n$ distinct or identical bins with all possible combinations.

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If n balls are thrown into k bins, what is the probability that every bin gets at least one ball?

If $n$ balls are thrown into $k$ bins (uniformly at random and independently), what is the probability that every bin gets at least one ball? i.e. If we write $X$ for the number of empty bins, what ...
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3answers
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How can I solve bins-and-balls problems?

Below is the problem that I wanted to solve When there are $m$ balls and $n$ bins, balls are thrown into bins where each ball is thrown into a bin uniformly at random. What is the expected number ...
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votes
3answers
238 views

Second pair of matching birthdays

The "birthday problem" is well-known and well-studied. There are many versions of it and many questions one might ask. For example, "how many people do we need in a room to obtain at least a 50% ...
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3answers
530 views

Expected max load with $n$ balls in $n$ bins?

If you throw $n$ balls into $n$ bins uniformly and independently at random, let $X$ be the number of balls in the bin with the largest number of balls in it. Is there an elementary way to ...
8
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1answer
154 views

How many expected people needed until 3 share a birthday?

I asked a somewhat related question recently and then became interested in this one: how many people are required, on average, until 3 share a birthday? More generally, if we have $M$ bins, what is ...
9
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3answers
382 views

Probability $k$ bins are non-empty

The following problem arises in the analysis of Bloom filters. Consider $m$ bins and $N=nk$ balls placed uniformly and independently at random into the bins. A query chooses $k$ bins uniformly and ...
8
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5answers
596 views

Probability that all bins contain strictly more than one ball?

Here's the problem I'm working on: Given that I'm distributing $N$ balls into $K$ bins, what is the probability that all bins contain at least two (strictly more than 1) balls? This seems like a very ...
7
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2answers
238 views

Find: The expected number of urns that are empty

A total of $n$ balls, numbered $1$ through $n$, are put into $n$ urns, also numbered $1$ through $n$ in such a way that ball $i$ is equally likely to go into any of the urns $1, 2, . . . , i$. Find ...
4
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1answer
257 views

Maximum load of a bin in the $n$ balls with weights into $m$ bins problem

$n$ balls, each with a weight $p_i$, are thrown into $m$ bins. Each bin is chosen with uniform probability. Prove or disprove that the expected value of the maximum load among the loads of bins is ...
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2answers
300 views

Throwing $k$ balls into $n$ bins.

I have the following question: Throwing $k$ balls into $n$ bins. What is the probability that exactly $z$ bins are not empty? I thought about something like: $$\Pr(z)=\frac{n! z^{k-z}}{n^k ...