# Tagged Questions

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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### Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
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### 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 ...
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There are $N$ buckets. Each second we add one new ball to a random bucket - so at $t=k$, there are a total of $k$ balls collectively in the buckets. At $t=1$, we expect that at least one bucket ...
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### Expected value of size of subset

Given a set $S$ such that $|S|=n$, A random item is chosen randomly from $S$, and being appended to a new set $T$. This process is being repeated $n$ times (with repetition), what is the expected ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Repeatedly Toss Balls into Bins

$n$ balls are randomly tossed into $m$ bins, each bin can hold $k$ balls. If a ball is tossed into a full bin (already has $k$ balls in it), it can be tossed repeatedly and randomly into the $m$ bins ...
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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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### Are “Balls in Bins” and “Stars and bars” the same?

I want to know that if Balls in Bins and Stars n Bars problems in combinatorics are similar? Can we reduce one into other? How? How can they be mapped to each other?
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### $m$ balls into $n$ urns

Assume that there are $m$ balls and $n$ urns with $m\gt n$. Each ball is thrown randomly and uniformly into urns. That is, each ball goes into each urn with probability $\dfrac1n$. What is the ...
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### 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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### Yet another balls and bins problem

If $p_n$ denote the probability that when $n$ balls are randomly put in $n$ bins then there is at least one bin with exactly one ball. Is there a simple (involving only little computation) reason for ...
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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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### $n$ balls are thrown randomly into $k$ bins - how many are empty?

A large number of variants of this question were already asked here, including these - one, two, which are close, but none seem to answer my question. Assume that $n$ balls are thrown randomly and ...
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### How many ways can you put 8 red, 6 green and 7 blue balls in 4 indistinguishable bins?

Assume all balls with the same color are indistinguishable. The order in which balls are put in a bin does not matter. No bins are allowed to have the same distribution of balls! For example, this ...
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### $m$ balls into $n$ already occupied urns

Assume that there are n urns, $k\in\{1,...,n-1\}$ of which have already at least one ball. $m$ new balls will be thrown into urns. Each of the $m$ balls is thrown randomly and uniformly into $n$ urns. ...
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### We throwing $m$ balls to $n$ cells…

We throwing $m$ balls to $n$ cells randomly... At each cell can be more then one ball, or (of course) it can still empty. What is the expectation of the empty cells? I'd like to get any help! Thank ...
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### Find the probability that the $4$th ball removed from the box is white

A box has $10$ balls, $6$ of which are black and $4$ of which are white. $3$ balls are removed from the box, their color unnoted. Find the probability that a fourth ball removed from the box is white, ...
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### Balls and bins conditioned on the number of non-empty bins

The expected number of occupied bins in the standard balls and bins problem (with $m$ balls into $N$ bins) is $N\left( {1 - {{\left( {1 - \frac{1}{N}} \right)}^m}} \right)$. How do I compute this ...
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### Picking a uniformly random subset of colored balls: moments of the discrepancy between colors?

Let $N \gg k\geq 1$, and $\gamma \in (0,1/2)$. I have a set of $N$ colored balls (3 colors in total), where $N$ is a multiple of $\gamma N$ are red; $\gamma N$ are blue; $(1-2\gamma)N$ are green. ...
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### Number of ways of distributing balls into boxes

I know that the formula for counting the number of ways in which $n$ indistinguishable balls can be distributed into $k$ distinguishable boxes is $$\binom{n + k -1}{n}$$ but I am having a hard time ...
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### Is there a fast, reasonably accurate estimator for multinomial PDF?

I am working on a balls in boxes kind of problem, where the probability of a ball ending up in a certain box varies by box, that is, each box has some probability P of getting any ball, all together ...
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### A Question on distribution numbers

This is a question from the book Combinatorics -a problem oriented approach which states: Q.1 Find the no. of distributions of a set of distinct balls into a set of distinct boxes, if no boxes can ...
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### Probability of having $k$ empty urns after putting $n$ balls into $n$ urns

Assume that there are $n$ balls (numbered from $1$ to $n$) and $n$ urns (numbered from $1$ to $n$). At the beginning no ball is placed in any urn. Balls are randomly thrown into urns: Each ball is ...
Given $N$ bins and an unlimited number of balls, to be assigned one ball per time unit to a uniformly random bin. What is the expected time to achieve at least 2 balls in every bin? From a previous ...