Questions tagged [balls-in-bins]

For problems on the distribution of $m$ distinct or identical balls into $n$ distinct or identical bins, optionally with restrictions.

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What is the probability of getting a red ball exactly $n-2$ times? [closed]

Problem An urn contains $3$ white balls and $5$ red balls. $n$ balls are drawn successively in such a way that at each drawing they’re still within the realm of possibilities, so $3^n$ drawings total. ...
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Possibilities of fixed point-free tupels

We define the sample space $\Omega:=\{\omega\in\{1,\dots n\}^n\mid \omega_i\neq\omega_j\text{ for all } 1\leq i<j\leq n\}$. We call a tupel $\omega\in\Omega$ fixed point free if for all $1\leq i\...
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Balls and bins lower bound with random algorithm

Assume we have $n$ bins and $B \sim Binom(nk,p)$ balls to distribute in an equitable. Furthermore, assume that each ball arrives one at a time and must be placed in a bin before the next arrives. ...
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Finding number of ways to put four balls in k bins

I'm trying to calculate how many ways we can put four indistinguishable balls in n indistinguishable bins, given that the maximum each bin can hold is r balls.
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Probability of putting $n$ balls into $k$ bins and having exactly $j$ bins empty?

What is the probability of randomly throwing $n$ balls into $k$ bins and having exactly $j$ bins without a single ball in it? There is a question with an answer that can be used to give the odds that ...
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1 vote
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Distributing particles into boxes

In our homework we had the following problem: Let's consider $n$- many particles. Each of them has the probability $\frac{1}{N}$ to land in one of $N$-many numbered boxes, where $N>n$. We select $...
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Ballot problem with tie

We have two candidates $c_1$ and $c_2$. At the end of a ballot each candidate receives $p$-many votes. What is the probability that $c_2$ has never been ahead of $c_1$ the whole process of vote ...
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ball and bins probability

If there are k bins and n balls are randomly thrown into the k bins (equal chance of every ball going into any bin) and then we randomly select a bin, what are the odds that the randomly selected bin ...
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Probability that fourth number is greater than first number

We have a bag which contains $49$ numbers. We draw $6$ numbers randomly without replacement. What is the probability that the fourth number is greater that the first one? My approach: First we define ...
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How many ways are there to put seven white and two black billiard balls in nine pockets? Some of the pockets may remain empty and the pockets are cons

How many ways are there to put seven white and two black billiard balls in nine pockets? Some of the pockets may remain empty and the pockets are considered indistinguishable. Well, my solution goes ...
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Distribution of $10$ different marbles to $6$ different color boxes

The question is: There are $6$ boxes, with each box a different color, and $10$ different marbles. The marbles are scattered randomly in the boxes. What is the probability that all marbles are in ...
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Classic Picking Balls from 2 Urns Question

The question involves 2 urns, URN1 and URN2. There are 18 balls of 4 variations in URN1: 3 Red Balls, 5 Orange Balls, 5 Yellow Balls and 5 Green Balls. There is 1 ball of unknown color in URN2, i.e. ...
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Distinct Objects into Distinct Bins, each bin has a different number of objects

I was curious about these types of distribution and combinatorics problems and going about solving them. Let's say there are 15 different fruits that are to be equally likely distributed among 6 ...
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Finding the number of ways of posting six letters in three letter boxes such that no letter box remains empty.

Find the number of ways of posting six letters in three letter boxes such that no letter box remains empty. This question already has answers here. But my doubt is in the following method. $$^6C_3\...
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Analysing Failure probability of random sampling with replacement

I am trying to understand the failure probability of random sampling with replacement. Here is one problem instance. Given a bag with M = 5 numbered balls (1, 2, 3, 4, and 5), from which the balls are ...
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Probability of selecting red balls from a bag

I am trying to solve this problem but getting stuck in the general case. Here is the problem statement: A bag contains a total of N balls. Out of which M balls are red, and N-M balls are blue. Here M &...
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Given $k$ balls and $n$ buckets where $k\geq n$. What is the expected value of number of balls in bucket $i$

Given $k$ balls and $n$ buckets where $k\geq n$. Each ball is thrown, and with probability $p$ it lands in one of $n$ buckets. Each bucket is equally likely. Let $X_i$ be a random variable represent ...
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1 vote
1 answer
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Combinatorics Problem distribute balls in boxes

Suppose 10 different red balls and 15 identical black balls will be distributed in 4 ballot boxes. Each urn must contain at least one red ball and the fourth urn must have at least 6 black balls. What ...
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How many ways can we put 2n white and n black balls into 3n different cells?

There are 3n different cells, 2n white balls, and n black balls. Each cell cannot be empty and must contain exactly one ball. How many ways can we put those balls into those cells? My solution is: Let'...
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1 answer
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How would you use Stirling numbers to partition a n unique objects, into m unique subsets, with minimum cardinality of 2 for each subset?

For example, how would one go about partitioning 10 numbered balls into 4 numbered boxes, such that each box contains at least 2 balls?
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3 votes
1 answer
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$2n+1$ numbered balls in box in arithmetic progression

We have $2n+1$ numbered balls in a box. Three balls are chosen randomly and simultaneously. Find the probability that the numbers are in arithmetic progression. My attempt to solve it: I have defined ...
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Distributing n identical balls into k distinct boxes when the number of boxes is greater than the number of balls and any box can have zero balls

Distributing n identical balls into k distinct boxes when the number of boxes is greater than the number of balls and any box can have zero balls. For eg distributing 4 balls in 7 boxes and any box ...
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1 vote
1 answer
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Number of distributions of balls in boxes

I have been thinking for this problem and I found it harder than I expected: Find de number of distributions of 1000 red balls, 1000 yellow balls and 1000 blue balls in 3 boxes with every box having ...
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2 answers
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In how many ways can 30 identical balls be distributed into 6 distinct boxes (numbered box 1, ... , box 6) where each box gets an odd number of balls? [closed]

In how many ways can $30$ identical balls be distributed into $6$ distinct boxes (numbered box 1, ... , box 6) where each box gets an odd number of balls? I have this problem at the moment and the ...
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Pairwise independent balls in bins: probability of an empty bin

There are $n$ balls. We put them into $2n$ bins uniformly and pairwise-independently, meaning $P(\text{ball }a \text{ in bin }B_i)=\frac{1}{2n}$ and $P(\text{ball }a \text{ in bin }B_i\text{ and }\...
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Different ways to put n distinguishable balls in k distinguishable boxes, when every box has to have an even amount of balls? [duplicate]

I've been learning about combinatorics, and up until now I have an okay understanding of generating functions and some basic situations like putting (in)distinguishable balls into (in)distinguishable ...
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A question about balls-in-bins problem. [closed]

Drop $n$ balls into $k$ bins in an independently random way. Let $X_i$ be the number of balls in the bin with label $i$. Let $s\geq1$ be some integer. Let $Y_s=\sum_{i=1}^kX_i^s$. It is clear that the ...
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combinatorics - distinct objects into identical cells - when order matter

I got the following question: how many options we have to divide N distinct objects into M identical cells when the order matter? to be honest, Im a bit baffeld about how "order" matter ...
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2 answers
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In how many ways we can place 9 different balls in 3 different boxes such that in every box at least 2 balls are placed?

In how many ways we can place $9$ different balls in $3$ different boxes such that in every box at least $2$ balls are placed? Approach 1: Let $x_1$, $x_2$, $x_3$ denote the number of balls in boxes $...
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2 votes
2 answers
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To find the number of ways to put $14$ identical balls into $4$ bins with the condition that no bin can hold more than $7$ balls.

To find the number of ways to put $14$ identical balls into $4$ bins with the condition that no bin can hold more than $7$ balls. I have tried the following: The total no of ways to distribute $14$ ...
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1 vote
1 answer
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Drawing balls from 3 urns, no replacement

I’ve been trying to figure out the solution of this problem for a while. Can someone help me? Three urns A, B and C contain respectively 3, 5 and 7 balls. In each urn there is a red ball, while the ...
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1 vote
1 answer
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How many ways we can load $100$ indistinguishable boxes into $3$ cars A,B,C,if every car can be loaded with $20$ to $40$ boxes?

I am trying to solve some exercises. Those exercises have to do with P(n,r) and C(n,r).There are exercise 1: How many ways we can load $100$ indistinguishable boxes into $3$ cars A,B,C,if every car ...
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2 answers
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Ways to divide $14$ numbered balls into $3$ non-empty groups

Fourteen numbered balls (i.e., $1$, $2$, $3$, $\ldots$, $14$) are divided in three groups randomly. Find the probability that sum of the numbers on the balls, in each group, is odd. There is this ...
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3 votes
2 answers
191 views

Distributing $r$ balls into $n$ cells. What is the probability that exactly $m$ cells contain exactly $k$ balls?

$r$ balls are randomly distributed into $n$ cells (the balls are indistinguishable). What is the probability that there is exactly $m$ cells that contains exactly $k$ balls (each one)? That is, the ...
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  • 475
4 votes
3 answers
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Colored ball problem with $2$ colors, but each ball is always replaced with $1$ color

I'm trying to find a solution to this problem but I'm having a hard time thinking through how to solve it: You have one bag with $N$ balls, each being red or white. You draw some number of balls, each ...
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8 votes
3 answers
510 views

I've clicked XKCD's "random" button k times and I've already seen all of them. What's the expected number of XKCD's I've seen?

This seems like a modification of the coupon collector's problem which can be stated as follows: There are $n$ coupons total to collect. Given that the past $k$ coupons seen I've already collected (...
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Expected fraction of bins with exactly k balls, and finite bins

I need someone to challenge my logic. This problem concerns balls in finite-capacity bins. There are similar problems but all I can find relate to throwing balls into bins and discarding those balls ...
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1 vote
0 answers
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Assigning balls to urns such that no urn contains more than one red ball

I have the following problem. Let's assume I have a set of urns $\mathcal{U}=\{u_1,\ldots,u_k\}$ and a set of balls $\mathcal{B}=\{b_1,\ldots,b_n\}$. Furthermore, $m$ of these balls are special (let's ...
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1 answer
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Expected value with balls and bins

Ten distinguishable balls are thrown into 100 distinguishable bins independently and uniformly at random. Let X be the number of bins with at least two balls. Compute E(X) and use Markov’s inequality ...
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5 votes
1 answer
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Distributing $10$ indistinguishable cars, $12$ indistinguishable balls, $14$ indistinguishable teddy bears to $3$ children, each have exactly $7$ toys

I have $10$ indistinguishable cars , $12$ indistinguishable balls, $14$ indistinguishable teddy bears.I want to distribute them to $3$ different children in a kindergarten such that each child will ...
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0 votes
1 answer
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Variance of balls in bins

I am trying to figure out which approach is appropriate for the next problem. I throw the ball into the bin with 30% probability of succes. If successful, I get 5 points and 0 if I fail. The task is ...
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1 vote
0 answers
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Placing balls in bins expectation

Suppose we have $n$ bins numbered $1, \ldots, n$ and $n$ balls numbered $1, \ldots, n$. We randomly place the balls in the bins, one per bin. What is the expected number of balls where the bin number ...
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2 votes
2 answers
75 views

Distribute n distinct objects into r alike boxes such that, there are at least 2 objects in a box.

The go-to approach for distributing n distinct objects into r alike boxes is S(n,r). (Stirling no. of the second kind). How do I make sure there are at least 2 objects in a box?
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0 votes
1 answer
51 views

Placing n balls into n buckets with one remaining.

You have n balls numbered 1,2,…,n that you are placing in n distinct buckets. In how many ways can you do this such that:(i) no bucket remains empty? (ii) exactly one bucket remains empty? $$\texttt{...
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2 answers
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Given N balls and M buckets(N > M), what is the probability that every bucket has at least one ball in it? [duplicate]

In reality, I want to find the probability that at least one bucket is empty. But I'm not sure whether it is easier to calculate that or do 1 - P(every bucket has a ball in it).
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4 votes
1 answer
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n black balls k white balls in M bins, what is the probability of selecting a black ball from any bin.

There are $n$ black balls and $k$ white, with $M$ bins. The process for filling the bins is as follows: Consider each of the $N=n+k$ balls, one at a time. For each ball, select 1 of the $M$ bins ...
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1 vote
1 answer
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Confusing in the solution of indistinguishable balls in distinct urns

This question was taken from Mathematical Statistics and Application $7$th edition page $85$ , question $2.181$ Suppose that n indistinguishable balls are to be arranged in N distinguishable boxes so ...
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1 vote
2 answers
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Balls in urns with different max capacity for each urn

I'm looking to figure out the formula for putting K balls in N urns, where each of the N urns has a specific max capacity. For example, Urn A has capacity for at most 5 balls Urn B has capacity for ...
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1 vote
1 answer
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Balls into bins with left and right bins always empty

N balls are sequentially and randomly allocated into M bins arranged in a circle. If a bin receives a ball, its left and right bins cannot receive any ball and are always set to be empty (new coming ...
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0 votes
2 answers
70 views

Find the number of ways in which 6 distinct balls can be placed in 5 distinct boxes such that each box has at least 1 ball

My approach :- I first assumed all balls to be similar in nature , so that would give me 5 ways to distribute the balls in the boxes , which will be 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 ...
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