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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Baseball related problem (balls and boxes)

Thanks in advance for any help! So I am trying to figure out if the number of hits an inning of baseball is random, or if hits tend to come in bunches. To do this, I'm just using a fairly small ...
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88 views

Number of ways to write a tuple of positive integers as a sum of tuples with certain constraints

There are $N$ boxes into which we put $mn$ balls in $m$ steps, where in each step we insert $n$ balls, each of which goes into a different box. In how many ways can we do this so that box $B_i$, $1 ...
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190 views

What is the “standard balls-in-boxes” argument?

I came across a question recently that I did not know how to do. It included a solution, and basically the question boiled down to how many positive integers are there with 10 or less digits such that ...
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57 views

Combinatorial object and formula for n distinguishable balls in k distinguishable bins?

Balls and bins have labels; n >= k. Bins must have 1 or more balls, but balls can only be assigned once. For now, I will say all balls need to be assigned. In reading Wikipedia on the twelvefold ...
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1answer
63 views

Expected value for the number of tries to draw the black ball from the bag

We have a bag with $4$ white balls and $1$ black ball. We are drawing balls without replacement. Find expected value for the number of tries to draw the black ball from the bag. Progress. The ...
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3answers
83 views

The probability that each child gets at least one chocolate is?

5 different chocolates are to be distributed among 4 children.The probability that each child gets at least one chocolate is ? Total number of ways is $4^5$.Got that.After that what to do?
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4answers
46 views

Finding the probability function

Jug contains $8$ balls, $3$ are red and $5$ are blue.pulling out balls with no return intill the first blue ball comes out $\underline{\color{red}\bullet\times3 ...
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2answers
30 views

Intuitively, how many balls are in the bag?

You have a bag that contains all white balls, though you have no idea how many. You reach into the bag and pull out 10 white balls. Then, with the help of a red marker, you mark those 10 white balls ...
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1answer
44 views

Expected number of different colored balls in the same bin

Let's say I have $K$ bins and big jar of balls containing $X$ balls. The jar actually contains $W$ white balls and $B$ black balls. Now I'll extract one ball from the jar and throw in one of the $K$ ...
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43 views

Balls and Bins [Raab 1998 proof]

I cannot work out the proof in one of the steps. The following is copied from the original paper “Balls into Bins” — A Simple and Tight Analysis: The case when $n\log n \ll m \leq n ...
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1answer
108 views

Number of ways to place $n$ balls in $m$ bins such that no two balls are placed in the same or adjacent bins?

Assume $m$ distinct bins that are placed on a circle. Therefore, each bin has two neighbors that are its adjacents bins. What is the number of ways that one can place $n$ indistinguishable balls in ...
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1answer
35 views

Balls and Bins- Heavy Loaded Case: A Tight Formula

I have $m=10^6$ balls and $n=13\cdot 10^3$ bins. I need to know how I can calculate maximum load for each bin. I'm aware of [1], but it's to loose (e.g. max_load: $e\cdot m/n$. And in practice using ...
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26 views

Selecting point of reference when counting

When arranging people (A, B, C, D, E) in seats (1,2,3,4,5), why isn't it same to count with reference to seats eg. (number of people can sit in seat number 1) * (number of people can sit in seat ...
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2answers
74 views

a generic ball and urn problem

Generic ball and urn problem: If two balls are randomly placed in three urns, what are the chances the second urn is occupied? In the generic ball and urn problem, since the balls aren't stated ...
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40 views

I need a help to understand a paper about Balls and Bins and bins max load

I must declare that I have asked a similar question before but I did not get any answer. I do need a concrete example and formula allowing me to determine maximum number of balls in a bin with ...
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1answer
49 views

Bayes theorem. Conditional ball draws

You have a white ball in your hand and there is a ball in a black box that has a probability of 50% of being white. You put the white ball you are holding in the box and shake up the box. a. ...
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2answers
71 views

A tricky conditional probability question involving recursion

I've been trying to answer this question for the past few days, and I'm absolutely stuck. Without further ado, here's the mystery: We are given a pair of boxes. There are n red balls in box number 1, ...
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Balls and Bins- Hash Table: a Concrete Example

My question is related to this: http://cs.stackexchange.com/questions/49027/balanced-allocation-hash-table-overflow-probability/49030#49030 In [1,2], it is said that if we throw $n$ balls into ...
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1answer
120 views

Probability that none of 3 tennis balls chosen at random have been used before

There are 15 tennis balls in a box, of which 9 have not previously been used. 3 of these balls are randomly chosen, played with and then returned to the box. later, another 3 balls are randomly chosen ...
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1answer
60 views

Balls into bins, probability that $k$ of the bins has at least 2 balls

$n$ balls are thrown uniformly randomly into $m$ bins, I'd like to calculate the probability that exactly $k$ bins have at least 2 balls. This is of course doable using the multinomial distribution, ...
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53 views

Labeled balls and urns. Finding the probability for a specific pairing.

I was reading a text and came across this problem. The wording is a little terse so I'm not clear on why the result is true. Given N balls and N urns, all of which are labeled. The balls are ...
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51 views

Balls into bins with repelling balls and bins, probability 1 or more of a specified subset of bins is empty?

I have $N$ sets of differently colored balls, each set a distinct color and each set consisting of 3 balls. There are 9 bins. Balls are tossed and will land into bins uniformly randomly, except if ...
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1answer
47 views

Drawing balls from one of two boxes based on a coin toss

Two boxes, the first contains 3 red & 2 white balls. The second box contains 4 red & 7 white balls. A fair die is tossed. If the outcome is divisible by 2, a ball is drawn from the first box, ...
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57 views

Balls and Bins when #balls < #bins

I did some search on Balls and Bins problem. But I haven't found out any bounds when the number of balls ($m$) is less than the number of bins($n$). I am aware of the cases when $m \geq n$ so will ...
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1answer
58 views

Small o-notation in probability [Raab 1998 proof]

I need a clarification about the notation used in the main theorem of the proof "Balls into Bins" - A Simple and Tight Analysis The theorem states that: Let $M$ be the random variable that counts ...
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204 views

Expected value of size of subset

Given a set $S$ such that $|S|=n$, A random item is chosen randomly from $S$, and being appended to a new set $T$. This process is being repeated $n$ times (with repetition), what is the expected ...
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3answers
225 views

Number of ways of distributing balls into boxes

I know that the formula for counting the number of ways in which $n$ indistinguishable balls can be distributed into $k$ distinguishable boxes is $$\binom{n + k -1}{n}$$ but I am having a hard time ...
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1answer
158 views

Total number of empty boxes

If you randomly put 18 balls into 10 boxes, what is the expected number of empty boxes? I tried the following: Let $X_{i} = {1}$ if i-th box is empty $X_{i} ={0}$ otherwise Then, let $Y$ be No. of ...
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1answer
142 views

Probability of events after a random ball was moved from one box to another

One box contains 3 white and 6 black balls. A second box contains 5 white and 4 black balls. One ball is chosen at random from the 1st box and put into the 2nd box. Then a ball is randomly ...
3
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1answer
55 views

Expected value of trials to obtain a red ball in a box of white balls.

I have a problem that involves a box containg N balls, one of which is red and the rest of which (N-1) are white. The question involves finding the expected value and variance for the number of trials ...
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39 views

Probability of having K cigarettes in one of the 2 packs of cigarettes

We have 2 packs of cigarettes, N cigarettes in each of the pack. We keep taking out a cigarette from those packs randomly, until one of the packs gets empty. How much is probability that the other ...
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4answers
166 views

Application of Pigeon-Hole Principle to balls in bins.

Given $n$ balls placed in $m$ boxes, prove that if $n < \frac{m(m-1)}{2}$ then at least two boxes have same number of balls in them.
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73 views

Bound the probability that the number of red balls in a random partition of a set of balls into 2 equal sized sets is close to the mean.

There are $2N$ unique balls, $2K$ of them are red, the rest are white (thus $2K \le 2N$). You divide the $2N$ balls randomly into two sets, each containing exactly $N$ elements. Is it possible to ...
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0answers
128 views

Partition counting problem with cap on pairwise intersection

Fix $T_1,\ldots T_m$ as pair-wise disjoint $k$-subsets of $\{1,2,\ldots,m\cdot k\}$, so that $|T_i|=k$ and $|T_i\cap T_\ell|=0$. For any $j\le k$, how many sets of the form $\{C_1,\ldots,C_m\}$ are ...
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The probability that exactly $k$ balls are in the first urn.

Suppose that $n$ balls are distributed randomly into $m$ urns. Determine the probability that exactly $k$ balls are in the first urn. Progress: I treated the balls as indistinguishable and the urns ...
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1answer
109 views

Jar and Ball Probability Distribution

If I have 8 jars, each jar contains 5 unique ball types. However, I know that I have 20 unique ball types out there. So, I have balls labelled from B1, B2, B3, ...B20 to put into 5 jars. Let's say ...
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1answer
53 views

Expected number of boxes with a blue ball [closed]

Suppose $N$ blue balls, $N$ red balls and $N$ yellow balls are placed in $N$ boxes such that each box has $3$ balls. What is the expected number of boxes with a blue ball? I know it's going to ...
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1answer
40 views

Probability of white ball appearance

There is a basket with balls. There are $M$ white balls and $N-M$ black balls. We take out all balls one after another. Find probability of $k$th ball to be white. Okay, let $A_k$ - be event of $k$th ...
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32 views

Given $d>8$ boxes and $n$ balls. . What is $P(A \cup B \cup C)$?

Given $d>8$ boxes and $n$ balls. Consider event $A$=boxes numbered $1,2,3,4$ receive 0 balls.$B=3,4,5,6$ receive 0 balls, $C=5,6,7,8$ receive zero balls. What is $P(A \cup B \cup C)$? ...
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1answer
65 views

Picking K counters out of K buckets containing NK counters, N of each different colour, up to N in each

This is a generalisation of a question that recently came up while solving a TopCoder problem. Suppose we have N blue counters, N red counters, N white counters, and so forth, K colours in total. We ...
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1answer
181 views

Number of ways to place $n$ balls in $k$ bins where just the first $r$ bins have less than $m$ balls

How many ways are there to distribute $n$ balls into $k$ bins where the first $r$ bins have less than $m$ balls (each) and the rest of the bins have more than $m$ (each)? Given this solution for ...
2
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1answer
198 views

Jee Main 2015 Question. Probabilty

If $12$ identical balls are to be placed in $3$ identical boxes, then the probability that one of the boxes contains exactly $3$ balls is: (1) $22 \times(\frac{1}{3})^{11}$ (2) $\frac{55}{3} \times ...
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1answer
210 views

Example of pairwise independent random process with expected max load $\sqrt{n}$. [closed]

Throw $n$ balls into $n$ bins. Each bin is selected uniformly at random but the process is only pairwise independent. Call the maximum number of balls in any bin the max load. Lemma 2 in these ...
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1answer
92 views

Balls and Bins not randomly

I'm trying to solve a problem similar to this one, I understand the answers of that problem, but my problem is Suppose N bins that are ordered and labeled from 0 to N-1. How many bins in mean do I ...
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1answer
137 views

Ways to distribute $400$ non distinct balls to $3$ bins such that in all of the bins there are more than $200$ balls or less than $100$ balls

How many ways there are to distribute $400$ non distinct balls to $3$ bins such that in all of the bins there are more than $200$ balls or less than $100$ balls. My attempt: Notice there's only ...
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142 views

Ways to distribute $25$ distinct balls to $40$ bins with constraints

How many ways there are to distribute $25$ distinct balls to $40$ bins such that: At least one bin will have more than one ball. One and only one bin (not predetermined) will hold exactly ...
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130 views

Possible combinations of N different balls in M identical bins with the same capacity L

For a distribution center I am interested in the number of possible combinations to put N different boxes (all the same size but different content) into M equal bins (containers) with capacity L, with ...
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1answer
79 views

How many different permutations?

Suppose I've n boxes and m different colored balls of different quantities.How many unique permutations can be obtained ? Example : n=2,m=2, with quantities ( A - 1 ball, B - 2 balls) Thus the ...
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1answer
193 views

Probability that P people will have N distinct birthdays

This question is rather difficult to describe clearly, so I will begin with an example. Suppose I have a 365 people in a room. The odds are very low that all these people have different birthdays. In ...
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Matching Question - a derivative of the birthday problem with two types

I am trying to calculate the expected number of matched trades given b buy orders and s sell orders, each on one of N underlying products. I think this can be cast as a balls-in-bins problem: If I ...