# 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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### Probability of 3rd ball being blue given at least 1 blue previously

I have a bin with $5$ blue and $10$ red balls. I take out $2$ and at least one is blue. I then take a 3rd. What is the probability that the 3rd ball is blue? I understand I likely need conditional ...
35 views

### Distribution of passengers into distinct cars

In four cars, 16 passengers randomly enter. What is the probability of the following events: a) 4 passengers enter the first car b) each car receives exactly 4 passengers I am trying to decide between ...
62 views

### Distinguishable objects into distinguishable boxes.

So here is the problem. We have $500$ fans of which $200$ Americans,$150$ Polish, $100$ Italians and $50$ French. They are about to travel to Paris for the start of the Olympic games with 5 different ...
1 vote
38 views

### Probability of no empty bins when $n$ balls go into $3$ bins

I'm a bit unsure what's wrong with my reasoning here: So we have $n$ balls, and want to place them into 3 bins, and we want the probability that $0$ bins are empty. Numerator: number of ways that $0$ ...
152 views

### Numbered Sets of Balls in Bins

For any prime power $N$, suppose we have $N^2+N+1$ indistinguishable bins, $N^2+N+1$ distinct people, and each person has $N+1$ indistinguishable balls. Each person throws all of their balls into a ...
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1 vote
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### $m$ distinct boxes and $n$ distinct balls probability problem [closed]

I have looked at similar questions, however I could not find an answer to my specific problem. The problem is the following: If I have $m$ distinct boxes and $n$ distinct balls, what is the ...
61 views

### In how many ways can 20212022 identical balls be arranged in 2022 different bins [with conditions]?

The problem goes like this: in how many ways can 20212022 identical balls be arranged in 2022 different bins so that the number of bins with exactly 4 balls is 3, with exactly 7 balls is 4 (2), and ...
119 views

### Balls are pulled and distributed randomly into the boxes. What is the probability that in no boxes there will be balls of the same color?

Full Question: The pool of 100 balls contains 20 pink balls, 10 yellow balls, 20 purple balls, 15 indigo balls and 35 green balls. We randomly pull 10 balls out of 100 and randomly distribute them ...
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1 vote
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### Finding the sample space of putting identical balls in identical boxes

My apologies for the vague title, I truly couldn't think of a better one. The question itself is quite tame compared to what's usually asked here, but is as follows: Suppose you have n balls and k ...
126 views

### 4 balls are placed into 4 buckets. What is the probability that exactly 1 bucket is empty?

Two approaches with different answers where exactly 1 bucket is empty: 1: If the balls are labeled and the buckets are labeled, the sample space has size 256. Then there are 4 choices for the bucket ...
1 vote
63 views

### Number of ways to put N indistiguishable balls into k indistinguishable boxes of limited size. [closed]

I've seen how to put N balls into k distinguishable boxes with no size limit (e.g. the multiplicity of an Einstein solid) where ${{N+k-1} \choose {k}}$ is the multiplicity. However, these boxes are ...
1 vote
113 views

### balls in bins with finite capacity and overflow: mean ball overflow

Say you have $M$ bins and $N$ balls, and each bin can only be filled with $C$ balls. At step zero, all bins are empty. Then at each step, one of the $M$ bins is chosen with equal probability, and a ...
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1 vote
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### Disperse distinct balls into identical bins with extra conditions

I asked this question today. I have problem in the second part: In how many ways can 4 red balls and 7 blue balls be arranged in 3 boxes where each box must contain at least 1 red ball and each box ...
1 vote
240 views

### Distributing balls into bins with some conditions

I have found a (In how many ways can 4 red balls and 7 blue balls be arranged in 3 boxes)(question). The question says: In how many ways can 4 red balls and 7 blue balls be arranged in 3 boxes where ...
1 vote
70 views

### Distribute balls in bins with extreme hard conditions

There are $6$ identical red balls, $9$ identical blue balls and $8$ identical yellow balls. Those balls will be distributed to $3$ distinct boxes such that the first box can contain at most $3$ balls, ...
61 views

### Why does $\frac {f(n,m-1)} {f(n,m)} = \frac {g(n,m-1)} {g(n,m)}$ here? (Balls and Bins)

Define $f(n,m)$ as returning the amount of ways to throw $n$ distinguishable balls into $m$ bins, and $g(n,m)$ as returning the amount of ways to throw $n$ indistinguishable balls into $m$ bins. I am ...
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### Why does order matter here for throwing $n$ balls into $n$ labelled bins?

I was studying the following practice problem: When throwing $n$ balls into $m$ distinguishable bins, what's the probability that bin $i$ has exactly $1$ ball in it? This scenario was modelled by ...
• 3,017
465 views

### Problem regarding coloring balls drawn from a bin

I'm not sure if this has been asked before, but here I have the following problem: There are $n$ indistinguishable red balls in a bin. Each "round," $k$ balls are randomly chosen from the ...
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### Probability that a ball drawn from a box, after drawing some balls, is white [duplicate]

Suppose we have a box of $n$ balls, from which, $n_1$ balls are white. we draw $k$ balls ($k<n$) from the box and we put them aside "without seeing them". What is the probability that the ...
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1 vote
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1 vote
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### Probability that at least one bin has all balls of the same color

There are $32$ balls in total, equally split into red, green, blue, and white colors ($8$ balls each). I also have $4$ bins. What is the probability that in at least one of the bins, all the balls are ...
• 13
354 views

### 20 two colored balls into 4 bins

We have $16$ red identical balls and $4$ black identical balls. We have $4$ bins and put all balls randomly into the bins such that each bin contains $5$ balls at the end. Each arrangement has the ...
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### balls and cells

For $n,k$ non-zero natural numbers we have $4n$ cells and $4k$ balls that are the same, and we need to fill the cells with the balls. in how many ways can we do that if we know that exist at least one ...
1 vote
26 views

### Cumulative distribution function, probability that $x$ urns or less are not empty

$k$ identical balls are thrown in $n$ urns at random. There may be more than one ball per urn. Let the random variable $W$ be the number of urns that are not empty. What would be the expression of the ...
1 vote
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### Balls in Bins with restrictions

The question is as follows: In how many ways can $30$ balls be placed into $10$ bins. The restriction is that any bin can only contain $0$, $2$, $3$ or $5$ balls. The balls are identical and the bins ...
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1 vote
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### Number of arrangements of balls in boxes

Given integer $0 < t < n$. There are $n$ boxes, labelled $1$ to $n$, is placed on a circle (which means, $n$ box and $1$ box is adjacent). Place a total of $n$ indistinguishable balls in the ...
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### Ball games: How to allocate $N$ balls into $M$ boxes so as to maximize the expected number of taken balls

Consider the following ball games, which looks like very intuitive and simple but I have tried for a long time. Assuming we have $M$ identical boxes and $N$ identical balls, we distribute these $N$ ...
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### balls in bins in a multinomial setting

Probability of observing $m_k$ bins over $m$ total bins that have $k$ balls inside each after $n$ identical balls have been launched in bins in a multinomial setting. More precisely, The occupancy ...
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### Draw balls from bin and add more balls after draws

A bin contains $r$-many red balls and $s$-many black balls. We draw randomly a ball and afterwards we return the ball into the bin and add $c$-many balls of the color we have previously drawn. We draw ...
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### The probability of having at least j bins, each has at least k balls

Assume that we have $m$ balls and $n$ bins. We throw the balls in bins randomly. Further, assume that $i<n$ and $0<\rho<1$. What is the probability of having $j$ bins $B^1, B^2, \dots, B^j$ ...
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1 vote
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### Density for number of collected golden balls from urns

$m$ golden balls are randomly allocated into $n$ binary urns, such that an urn contains either one or no ball ($m \leq n$). We select $k$ urns without replacement and collect the balls. Since we have ...
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### Safe bin capacity for throwing balls into bins without overflows

Suppose that we throw $n$ balls at random into $b$ bins, each with a capacity of $c$ balls. Given $n$ and $b$, what is a safe choice for $c$ such that there is at most a $q$ probability that any bin ...
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### Probability of picking multiple balls from a bag

I have a bag that contains numbered balls: Gray balls numbered 1-13 Green balls numbered 1-13 Blue balls numbered 1-12 Purple balls numbered 1-12 Yellow balls numbered 1-8 Each color of ball has a ...
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### In how many ways can $2n$ balls, two each of $n$ different colors, be placed in $n$ indistinguishable boxes so that two balls are placed in each box?

$\; \; \; \; \;$There are $2n$ balls with $n$ different colors, two for each color. (Balls of the same color are the same), and there are $n$ identical boxes. Now, we put all the balls into all the ...
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### Approximation to infinity number of balls thrown to bins, until one bin is full

Suppose that there are $n$ bins, each have capacity $C$. What is the expected number of balls can be thrown until one is full? This question is asked and answered here, but I am interested for an ...
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### how many ways are there to place identical balls in to urn if each urn has exactly same amount of balls?

I am currently struggling with the question in combinatorics: there are 24 balls, 12 of which are red and 12 are blue, and 3 urns. How many ways are there to distribute these balls independently of ...
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### Coupon Collector's problem, but coupons are distributed by a universal hash function

I am familiar with the Coupon Collector's problem. But what if instead coupons are distributed by a universal hash function? In other (more formal) words, if you hash set $S$ with function $h$ to ...
1 vote