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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What is the probability of picking $4$ balls out from the two bins?

Assume there are $2$ bins (bin $a$ and bin $b$ ), and each bin contains $n$ indistinguishable balls with $n>10$. The probability of picking $k$ ball from each bin is $p_k$, ($k\ge 0$). Then what is ...
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1answer
48 views

Combinatorics with balls and bins with constraint

I have $90$ identical balls to distribute among $64$ distinguishable bins. Each must get at least 1 ball, after that the distribution over the remaining $26$ does not matter. I know I first have to ...
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0answers
34 views

Calculating probabilities for distributing M balls in 4 bins scenario

I have the following scenario that I need to obtain the formula for it in order to program it: There are number of balls (M balls), and there are 4 different bins (Bin1, Bin2, Bin3, Bin4) that the ...
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2answers
46 views

probability that given n balls and 60 urns no urn has more than 1 ball

A carnival game is set up so that a ball put into play has an equal chance of landing in any of 60 different slots. The operator of the game allows you to choose any number of balls and put them all ...
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40 views

Balls and bins question with nlogn balls and n bins.

The question: Suppose we randomly drop nlogn balls into n bins. Give an upper bound on the expectation of the maximum number of balls in any bin. How would this be done? I believe the answer is well ...
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2answers
35 views

Inclusion/Exclusion Principle for indistinguishable balls into two different types of boxes

The problem asks me how to distribute n balls into h boxes where each must get at least 1 n and t boxes where each can be left empty What I have, and where I seem to have stalled is how to ...
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3answers
532 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 ...
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93 views

Balls and bins problem - expected number of balls needed to throw [closed]

Suppose we have n boxes and we start randomly and independently throwing balls into the boxes. (a) For a given box, what is the expected number of balls we need to throw before one of the balls lands ...
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1answer
186 views

Expectation of throwing $n$ balls into $n$ bins

Suppose we throw $n$ indistinguishable balls in $n$ bins at random. The throws are independent. What is the expected number of empty bins? What is the expected number of bins with one ball. Using ...
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54 views

How to estimate pigeonhole principle?

I was thinking about this after my professor mentioned the pigeonhole principle in class. Let's say we have $N$ items and $M$ containers. Here we assume $N > M$. We will randomly place each of the ...
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2answers
163 views

Probability in ball coloring

You have exactly $n^2$ balls each one of which can be colored in one of $n^2$ ways. That is total colors is $n^2$ but I am not saying all the $n^{2}$ balls are distinctly colored. However assume each ...
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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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2answers
133 views

Probability that $\frac{n}{2}$ bins are empty [close]

A Bloom filter of length $n$ was built. I have only the first $\frac{n}{2}$ bits of this filter. How will the false positive probability change? For the whole Bloom filter, the false positive ...
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1answer
116 views

Bins and balls model - filling first bins [close]

We have $n$ bins and $m$ balls. I want to compute the probability that in the first $k$ bins, $q$ of them will be non-empty. I can throw $m$ balls into $n$ bins in $n^m$ ways. Using Stirling ...
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1answer
137 views

Ball and bins & Coupon collector problem

If you know a coupon collector problem, you will know what I am talking about. But if you are not familiar with I will try to explain what is the coupon collector problem. I have $n$ bins. I throw ...
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2answers
101 views

How many ways to $22$ balls in $5$ boxes problem

How many ways are there to put $22$ identical balls into $5$ boxes, with each box has at least $2$ balls? The answer is $16$ choose $4$, i.e., $_{16}C_4$, but can someone explain it to me?
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2answers
54 views

Probability and conditional?

A bag contains four red balls and four white balls. A ball is selected at random, removed and replaced by a ball of the opposite colour. A second ball is then selected at random. Calculate the ...
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1answer
51 views

probability of specific number of balls in an urn

I have $n$ balls, I distribute them between 2 urns. Each ball goes to urn 1 with probability $p$ and urn 2 with probability $1-p$. What is the probability that there are $l$ balls in urn 1 when all ...
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1answer
354 views

Distribution of distinct balls in identical boxes

how can I derive a formula for the number of distributions of $n$ different balls in $k$ identical boxes. Where $\mathbf{empty\ box}$ is allowed. I know this is equivalent to finding the number of ...
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0answers
96 views

The probability of two balls going into the same bin

Suppose I have n bins and n balls with label on them. What would be the probability that I throw the i th ball into the same bin as the j th ball? Is it just 1/n because there is a probability of 1/n ...
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1answer
66 views

Balls and bins with 2 balls and 2 bins

I'm trying to understand a proof in a paper I'm reading. It relies on a balls and bins problem. Here is what I'm trying to figure out: We want the maximum number of balls in a bin. We have 2 balls ...
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1answer
345 views

Number of ways to distribute identical balls into identical bins

I have a question which I expected to be quite famous and common, yet I haven't found much... How many ways are there to distribute $k$ identical balls into $n$ identical bins? For example $(k, ...
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1answer
451 views

Balls and bins probability problem

$k = \sqrt n$ balls are thrown into $n$ bins. The bins are standing in a row and numbered from 1 to $n$. What is the probability that there are no two balls in the same bin or in adjacent bins??? In ...
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60 views

Maximum bin load for $\alpha n$ balls into $n$ bins

In a paper I am reading the author writes: A standard result concerning balls and bins shows that if we throw at least $\alpha n$ balls into at most $n$ bins, then the maximum bin load is ...
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2answers
133 views

A bin and balls problem

Throw 7 balls into 7 bins. Given there are exactly 2 empty boxes, find the probability that 1 bin contains 3 balls, and thus the other 4 bins contain 1 ball each. I know I could use a multinomial ...
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1answer
1k views

Distribute distinct objects in identical boxes

Number of ways to distribute $6$ distinct objects to $3$ identical boxes such that each box should have atleast one? $\mathbf {Is\ there\ any\ standard\ formula\ for\ these\ sums}$, as we have for ...
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1answer
111 views

Bounds for 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 a nice elementary way to ...
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1answer
63 views

Expected number in b bins with max k balls each, balls in bins question

Suppose we have b bins and each bin could have max k balls. We have sufficient balls and what is the expected number if we need to full all bins? I know the answer when k = 1, the expected number is b ...
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1answer
253 views

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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1answer
124 views

$n$ balls into $k$ baskets, $n \geq k$, no empty baskets

I have $n$ balls and throw them into $k$ baskets. None of $k$ baskets should be empty. Which means each basket has at least one ball. Balls and baskets are not distinguishable. What is the number ...
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1answer
175 views

Average number of bins occupied when throwing $n$ balls into $N$ bins

There are $n$ balls and $N$ bins. At each time, a ball is thrown in one bin of $N$ bins at random. This repeats n times. So that in total $n$ balls are thrown into bins. The question is, on average, ...
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100 views

Urn problem: replacing white balls with black once selected

I'm trying to find the probability of an outcome where, using the traditional example, white balls are replaced by black balls once selected. Initially I have $n$ white balls and $\mu$ samples. I ...
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121 views

Modified balls and bins

Assume that we have $n$ balls and $k$ bins. The $n$ balls are are divided into $M$ sets, $\left\{ {{m_i}} \right\}_{i = 1}^M$, where $\sum\limits_{i = 1}^M {\left| {{m_i}} \right|} = n$, $|m_i| \le ...
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1answer
248 views

What is the expected number of days in a year in which exactly $k$ people in a group of $n$ people have been born?

There is a group of $n$ people and we must find the average number of days that in each of them exactly $k$ people are born ($k$ and $n$ are given). This question assumes that a year has $365$ days, ...
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1answer
119 views

balls and bins - the probability of reaching the exact expected value

Suppose I throw $n$ balls to $m$ bins (randomly and independently). The expected value of number of balls in each bin is $n/m$. How can I tell (or bound) the probability that every bin has exactly ...
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1answer
317 views

Hyper Birthday Paradox?

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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2answers
193 views

Probability of number of unique numbers in $37$ Roulette Wheel spins.

I was wondering if someone could help me answer the following question: Calculate the probability that in $37 $ consecutive spins of a roulette wheel (using an European Roulette wheel) you will have ...
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1answer
99 views

In how many ways I can put $2$ red balls and $3$ green balls in $5$ boxes?

I have $5(N)$ boxes and some balls. Here's the description: Red Balls $= 2 (k1 = 2)$ Green Balls $= 3 (k2 = 3)$ ...
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1answer
211 views

Probability - Balls and Buckets; variance question

I've been working on this problem for a while and its giving me no end of trouble! The question is this: Suppose we have 2k buckets, numbered 1 through 2k. We throw x black balls and y white balls, at ...
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110 views

Yet another balls and boxes problem; minimum number of throws so that we have no empty boxes.

I managed to figure out how many empty boxes will be left given n amount of throws, just having a hard time figuring out the minimum number of throws necessary so that we have no empty boxes. Would it ...
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1answer
155 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 ...
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0answers
89 views

$m$ 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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1answer
210 views

$m$ balls into $n$ urns

Assume that there are $m$ balls and $n$ urns with $m>n$. Each ball is thrown randomly and uniformly into urns. That is, each ball goes into each urn with probability 1/n. What is the probability ...
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1answer
201 views

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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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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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 ...
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The distribution of balls in bins as a function of the number of balls [duplicate]

If I toss $m$ balls into $N$ bins, as a function of $m$, what distribution of balls and bins should I expect as a function of $m$? What would be the mean largest number of balls in a particular bin?
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2answers
240 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 ...
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4answers
207 views

$n$ balls into $n+1$ urns (with one special urn)

Assume that there are $n$ balls numbered from $1,2,\ldots,n$ and $n+1$ urns, numbered as $0,1,\ldots,n$ Throw each ball randomly into one of $n$ urns: urn 1, ...
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1answer
265 views

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 ...