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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 ...
user1352516's user avatar
0 votes
2 answers
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 ...
mario's user avatar
  • 3
0 votes
0 answers
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 ...
Sergio Guri's user avatar
1 vote
1 answer
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 $0$ ...
Mario Diw's user avatar
2 votes
1 answer
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 ...
Evan Semet's user avatar
1 vote
1 answer
55 views

Putting $k$ balls in n urns, which are grouped into $r$ sets.

Suppose I have $k$ balls, $n$ urns, and the urns are grouped into $2r+1$ sets where each set contains $\frac{n}{2r+1}$ urns. Assume that $n$ is a multiple of $2r+1$. (The choice of $2r+1$ rather than $...
Hans Schmuber's user avatar
0 votes
2 answers
53 views

No adjacent empty boxes with PIE

*PIE = inclusion-exclusion How many ways are there to distribute 10 balls into 5 distinct boxes such that no two adjacent boxes are empty? Note: the same question statement. I believe the question ...
zxayn's user avatar
  • 127
0 votes
2 answers
55 views

number of possibilities to distribute k balls in 2n boxes with condition

I'm trying to find the number of possibilities to distribute $k$ balls to $2n$ boxes such that for every $i$ between $1$ and $n$ the sum of the balls in the $i$ box and the $n+i$ box isn't equal to $6$...
user1188938's user avatar
2 votes
1 answer
75 views

In how many ways it is possible to take out balls from the basket , such that will take out at least one from each color?

The question: In a basket there are $20$ black balls, $15$ white balls and $18$ red balls. All balls with the same color are identical to each other. In how many ways it is possible to take out balls ...
User33975329257439645's user avatar
2 votes
3 answers
155 views

Number of ways of distributing white and black balls into bowls

This is a self-answer question that repeats this posted question that the original poster deleted themself. I regard the math problem as interesting enough to warrant posting. I will be very ...
user2661923's user avatar
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1 vote
2 answers
403 views

Probability of empty bin, where the number of balls is based on another game...

Came across this interesting question: We are given $j$ balls. We put them into bins labelled $1$, $2$, and $3$ at random. We want to calculate P(all bins have at least a ball). The approach I'm using ...
Eric Aldrin's user avatar
0 votes
1 answer
77 views

Chance that a group of $n$ people collectively have a missing birthday.

Suppose there is a city with $n$ residents, what is the probability that there is some day in the year at which none of them have their birthday? In other words, suppose we divide $n$ balls into $k=...
jorisperrenet's user avatar
2 votes
2 answers
77 views

$m$ 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 ...
Mathenthusiastic's user avatar
2 votes
1 answer
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 ...
sisyphusgottired's user avatar
3 votes
1 answer
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 ...
Luoencz's user avatar
  • 31
1 vote
1 answer
133 views

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 ...
Magniquick's user avatar
0 votes
1 answer
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 ...
Zach Smuckler's user avatar
1 vote
0 answers
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 ...
Hurricane's user avatar
1 vote
1 answer
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 ...
user3433489's user avatar
1 vote
0 answers
60 views

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 ...
user avatar
1 vote
2 answers
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 ...
user avatar
1 vote
0 answers
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, ...
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0 votes
1 answer
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 ...
Princess Mia's user avatar
  • 3,017
0 votes
3 answers
153 views

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 ...
Princess Mia's user avatar
  • 3,017
6 votes
3 answers
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 ...
Aiden Chow's user avatar
  • 2,848
0 votes
0 answers
18 views

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 ...
Fish_n_Chips's user avatar
1 vote
1 answer
121 views

N distinct balls in K distinct boxes with restrictions

How many ways are there to put N distinct balls in K distinct boxes such that each box has a max of C balls? Ordering of balls within boxes is not relevant. For instance, the ways you can distribute $...
pranjaln's user avatar
3 votes
1 answer
105 views

Number of ways to distribute $n$ items into bins, in decreasing order

I want to put $n$ items into arbitrarily many bins (equivalently, $n$ bins) such that each bin, has at most, as many items as the previous bin. The items are identical, so we only distinguish ...
Sebastián Mestre's user avatar
0 votes
0 answers
24 views

Estimate total balls based on the ball count in a bin

I am struggling to formalise the following problem. We randomly spread $m$ balls into $n$ bins. $m$ is unknown, but $n$ is known in advance. Now, we sample the first bin (or any bin) and see it has $k$...
Roman's user avatar
  • 133
1 vote
1 answer
56 views

Asymptotic growth of number of ways to distribute $10$ balls into boxes

What is the polynomial asimptotic bound of $ (a_n)_{n=10}^\infty$, where $a_n$ represents the number of ways to distribute $10$ identical balls into $n$ non-identical boxes, such that there are no two ...
Robert's user avatar
  • 597
0 votes
0 answers
65 views

Chance that all $k$ bins are filled by $n≥k$ balls without using PIE

Find the probability that all $k$ bins are filled, if each of $n≥k$ balls are randomly placed into one of the $k$ slots. I have solved the above question using PIE where I take the complement ($1$ - $...
as1092's user avatar
  • 13
1 vote
1 answer
75 views

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 ...
as1092's user avatar
  • 13
4 votes
4 answers
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 ...
Philipp's user avatar
  • 4,564
2 votes
1 answer
153 views

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 ...
yahel amity's user avatar
1 vote
0 answers
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 ...
Adrien Lessard's user avatar
1 vote
1 answer
47 views

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 ...
Triple_1's user avatar
  • 117
1 vote
0 answers
54 views

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 ...
atzlt's user avatar
  • 562
3 votes
0 answers
137 views

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 $N$ ...
koko's user avatar
  • 69
0 votes
0 answers
120 views

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 ...
FabIO's user avatar
  • 3
0 votes
2 answers
45 views

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 ...
Philipp's user avatar
  • 4,564
0 votes
0 answers
34 views

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 $B^1, B^2, \dots, B^j$ ...
Hasan Heydari's user avatar
1 vote
2 answers
80 views

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 ...
Pontus Hultkrantz's user avatar
0 votes
0 answers
42 views

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 ...
orlp's user avatar
  • 10.5k
0 votes
0 answers
73 views

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 ...
Matt Ryan's user avatar
0 votes
0 answers
129 views

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 ...
ame's user avatar
  • 11
0 votes
0 answers
59 views

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 ...
user3563894's user avatar
0 votes
3 answers
288 views

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 ...
laguna's user avatar
  • 169
0 votes
0 answers
41 views

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 ...
Gustav's user avatar
  • 1
1 vote
0 answers
67 views

m distinguishable bins, n distinguishable balls: probability of any bin containing *exactly* k balls.

Assume n distinguishable balls are independently and uniformly distributed among m distinguishable bins. What is the probability that at least one bin gets exactly k balls? This problem generalizes a ...
user1185597's user avatar
0 votes
0 answers
38 views

How many distributions can we make putting balls into boxes with a maximum capacity in box term [duplicate]

I need to distribute 100 similar balls in 10 distinct boxes but the maximum balls in one box can be 20. x1+x2+...+x10 = 100 xi ≤ 20 for every 1 ≤ i ≤ 10. how do I continue from here?
Husam's user avatar
  • 35

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