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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3
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2answers
41 views

Probability of at least $2$ balls in random bin when distributing $K$ identical balls to $N$ distinct bins?

If we distribute $k$ identical balls to $n$ distinct bins, what is the probability a randomly selected bin will contain at least 2 balls, assuming all balls have a uniform probability of being placed ...
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0answers
33 views

Balls and bins - average vs best case scenario

Assume we have $3$ bins: bins $L_1, L_2, L_3$, in each bin, there are $3$ balls. Alice is picking one ball from each bin at random. The balls can be either white or black. Alice does not know how many ...
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41 views

Number of ways to put 4 balls ( white and black ) to 3 urns?

The urns are non-distinct. The balls are non-distinct ( there aren't numbers on it, but every ball have one color, so they are a little distinct - there is difference between black and white ball ). I ...
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12 views

A “distributing redundantly” variation of the balls into bins problem.

We throw $n$ balls into $m$ bins, where $n \gg m$. If we repeat the experiment $s$ times, what will be the optimal value of $s$ so that any pair of such balls do not collide in at least one of the $s$ ...
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1answer
102 views

$n$ balls into $n$ urns until there is no empty urn.

We have $n$ balls and $n$ urns. Step 1. Throw $n$ balls into $n$ urns randomly. Check each urn and if there is more than one ball in an urn, choose only one and keep in the urn and remove all the ...
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0answers
47 views

Two types of $n$ Balls into $k$ Boxes

There are $n$ balls to distribute into $n/k$ boxes (each containing $k$ balls). $x$ red balls and the rest white balls. ( $x < (n-k)/2$) What is the probability/bound for having at least one box ...
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2answers
22 views

Expected value of pulling blue marble from sack on nth pull

I am doing a refresher for a work course and I'm a little stuck the question asks the following You have a sack with 7 marbles inside 4 Blue and 3 Green You draw all the marbles (without replacing ...
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1answer
25 views

Calculate empty labeled boxes in labeled boxes-balls problem

The problem is to count the number of times that K boxes are empty when n labeled balls are distributed into m labeled boxes. So, if we have n balls and m boxes, we have $m^{n}$ possibilities. I would ...
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1answer
59 views

We put $15$ balls into $5$ boxes. No box can contain more than $4$ balls. Prove that there are at least $3$ boxes, containing $3$ or more balls.

We put $15$ balls into $5$ boxes. No box can contain more than $4$ balls. Prove that there are at least $3$ boxes, containing $3$ or more balls. I know this problem is crying to be solved by the ...
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1answer
20 views

Balls in a box, volume off smaller balls an bigger balls, will it change with same weight?

If I have $10$ pounds of stainless steel balls of $2$ inch, then together they have a (pack)volume. Will the (pack)volume change if I switch to $10$ pounds of stainless steel $4 $ inch balls? (I think ...
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2answers
51 views

Variant of balls bin problem

I am stuck at the following variant of the balls bin problem. The task is to count the number of ways of placing $n$ identical balls into $k$ ordered bins under the constraint: the bin number $i$ must ...
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2answers
39 views

Balls in boxes - proof correction

I have been thinking about the following problem: How many ways can $50$ different balls be placed into $50$ different sized boxes if exactly $2$ boxes have to remain empty? My approach was the ...
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1answer
60 views

Calculating probability of getting at least 3 red ball drawn from a bag having 6 red ball,5 blue,4 green in 5th attempt. [closed]

There are 6 red balls,5 blue balls,and 4 green balls in a bag. What is the probability of drawing at least 3 red ball in 5th attempt. Assume balls are drawn with replacement I tried to use binomial ...
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1answer
38 views

How to find this probability with bags of balls?

I have two bags with different color of balls in it. In bag 1 there are 1 white ball and 3 black balls and in bag 2 there are 1 white ball and 2 black balls. I wanna know what is the probability for ...
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1answer
68 views

Every ball in different urn

We've got $n$ urns, I have to show that for some $c>0$ there are $c \sqrt{n}$ balls, that after putting them inside urns, probability that all the urns have at most 1 ball is equal or lower than $\...
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0answers
34 views

Counting methods

I would like to ask, if my intuitive understanding of how to count is correct. Please elaborate or correct me, if the below interpretation is not sound. Should constructing a $r$-sample (unordered set)...
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1answer
50 views

Generating function for distributing $k$ indistinguishable elements into $n$ distinguishable bins such that each bin has even number of elements.

Generating function for distributing $k$ indistinguishable elements into $n$ distinguishable bins such that each bin has even number of elements. So for the bins to have exactly even number of ...
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0answers
19 views

How many ways are there to arrange n balls into m numbered bins, if each ball is placed in a separate bin and balls are indistinguishable?

How many ways are there to arrange n balls into m numbered bins given that: each ball is placed in a separate bin balls are indistinguishable?
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2answers
37 views

Colored balls probability - color of remaining balls [closed]

A bowl contains 5 white and 8 black balls. We proceed to pick out balls one by one (without putting any back in the bowl) until only same-color balls remain in. What should I think to calculate the ...
2
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0answers
350 views

Finding “redundant” or “covered” vectors in a randomly sampled collection.

Fix integers $k,m,t > 0$. Say that I sample $w_1, \ldots, w_t$ uniformly and independently from $\{1,2,\ldots,m\}^k$. For each $i \in \{1,2,\ldots,t\}$, let $S_i = \{w_1,\ldots,w_t\}\setminus\{w_i\...
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1answer
68 views

In how many ways can I distribute $100$ identical balls into $6$ different boxes?

In how many ways can I distribute $100$ identical balls into $6$ different boxes so that no box is left empty and every box contains even number of elements? Firstly, I placed $2$ balls in every box, ...
2
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1answer
71 views

Balls into bins question. Where is the mistake?

I am not able to figure out a loophole in my argument related to popular "Balls into bins" problem. Suppose there are $m$ balls and $n$ bins. The $m$ balls are thrown independently and ...
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1answer
43 views

Probability of balls entering boxes

I had some troubles with the following exercise: Randomly throw 5 balls into one of the 4 boxes. What is the probability that no boxes are empty? I think the answer is P(A) = 4 * (5!/1!*1!*1!*2!) * (1/...
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3answers
98 views

Number of ways to distribute balls in three out of five boxes

Given 10 distinguishable balls and 5 distinguishable boxes (such that the boxes can be numbered 1 through 5): How many ways are there to distribute the balls so that all the balls are strictly in the ...
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0answers
30 views

Help in counting the number of distributions, the random assignment of $r=7$ balls to $n=7$ cells

I am a beginner in Probability Theory. Upon reading and solving few problems from the first five chapters from Feller's - Introduction to Probability theory and applications book, I know that, every ...
2
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1answer
50 views

$8$ distinct balls are randomly distributed among $4$ boxes. What is the probability that each box has exactly two balls?

I made up an question in my head as to probability but i am not sure about the solution. Question: There are $8$ distict balls and $4$ distinct boxes.An individual distributes the balls into boxes ...
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1answer
42 views

Isomorphism Between Counting Functions and Drawing Balls

I have a question regarding associations between different formulations of counting problems. My fundamental conception of such problems is as quantities of functions counted via the twelvefold way, ...
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1answer
24 views

Given fixed number of bins with equal capacity and variable number of items how many bins will I need?

I'm slightly embarrassed to ask this question as I feel that the answer might be exceedingly simple but I sat down to think about it and can't seem come up with a formula. I have a fixed number of ...
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0answers
31 views

Lower bound for maximum load in $m$ balls into $n$ bins problem ($m \neq n$)

The lower bound for maximum load when randomly placing $n$ balls into $n$ bins is known to be $\frac{ln(n)}{ln(ln(n))}$ and this lower bound fails to hold only with probability $\leq \frac{1}{n}$ for $...
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2answers
52 views

A box has 100 different balls. No of Balls that can be selected = 1 to 100. How many ways can we select?

A box have 100 balls. Each one is different. One boy can take atleast 1, atmost 100 balls. How many WAYS can he take balls ? ( clarification: Taking ball 1 then ball 2 = Taking ball 2 then ball 1 )
3
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1answer
127 views

Distribution of the maximum number of collisions

Given $n$ bins and $m$ balls, throw each ball into a bin that is chosen uniformly at random. Each throw is independent. What is the distribution of the maximum number of collisions (i.e. maximum ...
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0answers
32 views

How many balls spill in expectation? (balls and bins)

Consider throwing $n$ balls uniformly into $k$ bins, each of capacity $c$. Denoting by $X_i$ the number of balls thrown to bin $i$, I'm interested in understanding the quantity $$S_{n,k,c}=\sum_{i=1}^...
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0answers
38 views

Balls and bins: Probability of picking the same color given different numbers of selected balls

A bin has $n$ balls. $b$ balls are black and $w$ balls are white. I pick randomly $k$ balls, where $k$ can be $1$, $2$ or $3$ with equal probability. What is the probability that all selected balls ...
4
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1answer
95 views

Distribution of highest numbered occupied box and number of balls therein?

Say we have $N$ boxes, numbered $1,2,...,N$. We select $K$ numbers uniformly from $1,2,...,N$, and for each result we place one ball into the correspondingly numbered box. We then select $L$ numbers ...
3
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1answer
72 views

expected number of balls needed to throw until one of the bins becomes full

Consider the problem of throwing balls uniformly at random into $q$ bins, each has capacity $q$. What is the expected number of balls we need to throw until one of the bins becomes full? Clearly the ...
2
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1answer
58 views

What is wrong with my approach for CLRS 5.4-6 : Given n balls and n bins,find expected number of empty bins?

I am trying to find expected number of empty bins after n balls are tossed into n bins. And each toss is independent and equally likely to end up in any bin. Below is my approach. My indicator ...
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0answers
56 views

Finding expected indegree of a node and expected number of nodes with indegree=0

With a single node $v_1$, at every step $t$ with $1 < t ≤ n$, a single node $v_t$ is added to the graph, together with a single outgoing edge; the destination of the edge is a node chosen uniformly ...
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1answer
76 views

Show that when $n$ balls are thrown into $n^3$ cells, the chance that (at least) two balls will collide goes to zero as $n\to \infty$ [closed]

Show that when $n$ balls are thrown into $n^3$ cells, the chance that (at least) two balls will collide goes to zero, as a function of $n$: $$\lim_{n\to \infty }P(\exists\, x,y:x,y\text{ collide})=0$...
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1answer
54 views

Why probabilies are not adding up to one? [closed]

I want to find the probability of certain cases of picking 4 cards from 12 cards. Let's say the type of cards are A, B, and C, and there are 4 duplications of each type. The set of cards S={4.A,4.B,4....
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0answers
28 views

N Balls in two urns

Let A and B tow urns containing both N balls numbered from $1$ to $N$. 1) We assume to extract casually one ball from A and one ball from B. What is the probability that they have the same number? $...
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0answers
34 views

maximum load of balls and bins when $m \neq n$

When $m=n$, there is a classical result such that the maximum load would be $\theta(\frac{\ln n}{\ln \ln n})$. However, what happens when $m \neq n$? Say if $m = n^{0.5}$ or $m=n^{0.9}$, how can we ...
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3answers
45 views

Distributing balls into distinct boxes

So basically my question is this: I have $30$ non distinct balls that I want to put inside $4$ distinct boxes. For every $1\le i\le 4$, the $"i"$ indexed box must at least have $i$ balls and at max $...
1
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1answer
74 views

Combinatorics - how many ways to divide balls in two groups

Suppose I have: 8 black balls 3 white balls 5 blue balls how many ways there are to divide those balls into two different groups (note that there is no need to divide into two groups with even ...
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0answers
23 views

Placing balls of different colours into bins

I'm trying to prove the following statement: Suppose that you have 25 balls to place into five different bins. 11 of the balls are red, while the other 14 are blue. Prove that no matter how the ...
2
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1answer
85 views

N*n*m distinguishable balls with m different colors

I recently asked the following question which is resolved: n*m distinguishable balls with m different colors, the probability of randomly choosing k balls containing all balls from at least 2 ...
3
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2answers
121 views

n*m distinguishable balls with m different colors, the probability of randomly choosing k balls containing all balls from at least 2 different colors

Assume we have m groups of n balls and the balls in the same group have the same color. So there are m*n balls in total. Now, suppose we randomly choose k>(2*n) balls from the set of m*n balls. How ...
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0answers
25 views

Drawing balls out of two boxes [duplicate]

Suppose that on each turn it is equally likely to draw a ball out of box $A$ or box $B$. Each box contains $N$ balls. If you choose a box to draw a ball and notice that the box is empty then the round ...
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1answer
68 views

Estimate number of balls by picking a random bin

This is a tweak to the standard balls and bins problem where we (usually) come up with bounds on the max load or empty bins. I am interested in estimating $M$ when $M$ balls are (uniformly) thrown ...
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0answers
30 views

Balls in two sets of distinguishable boxes

I'm dealing with a physics problem right now that essentially boils down to a ball and box problem. I know that the distribution of N indistinguishable balls into k distinguishable boxes is given by ...
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2answers
109 views

Balls in bins with limited capacity

There are $6$ bins and $15$ identical balls. The first $5$ bins can only contain $3$ balls whereas the last bin can contain $1$ ball. When a bin is full, no balls can be thrown into that bin(if a ball ...

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