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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28 views

Probability of drawing a run of a specific color from an urn with two colors of balls

I was sent a puzzle involving an urn with 128 white balls and 288 black. If the balls are drawn without replacement until the urn is exhausted, what is the probability that a sequence of 10 or more ...
31
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3answers
6k views

Making 400k random choices from 400k samples seems to always end up with 63% distinct choices, why?

I have a very simple simulation program, the sequence is: Create an array of 400k elements Use a PRNG to pick an index, and mark the element (repeat 400k times) Count number of marked elements. An ...
0
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0answers
61 views

Balls and Boxes [closed]

How many ways are there to put 6 balls in 3 boxes if: a)the balls are not distinguishable and neither are the boxes? b)the balls are not distinguishable but the boxes are? c)the balls are ...
0
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2answers
57 views

How many positive integer solutions are there to the inequality $x_1+x_2+…+x_r\le n$?

The original problem is there are $r$ identical boxes and $n$ identical balls. Every box is nonempty. Then how many ways of putting balls in boxes? It is equivalent to the problem of finding ...
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1answer
59 views

Is there a fast, reasonably accurate estimator for multinomial PDF?

I am working on a balls in boxes kind of problem, where the probability of a ball ending up in a certain box varies by box, that is, each box has some probability P of getting any ball, all together ...
2
votes
3answers
53 views

Find the probability that the $4$th ball removed from the box is white

A box has $10$ balls, $6$ of which are black and $4$ of which are white. $3$ balls are removed from the box, their color unnoted. Find the probability that a fourth ball removed from the box is ...
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1answer
43 views

Expectation time for at least 2 balls in each bin

Given $N$ bins and an unlimited number of balls, to be assigned one ball per time unit to a uniformly random bin. What is the expected time to achieve at least 2 balls in every bin? From a previous ...
0
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1answer
86 views

$N$ bins, $m$ balls: Probability of any bin containing *exactly* $k$ balls.

This is an extension to question 821984: Given $N$ balls independently distributed randomly among $m$ bins, what is the probability that at least one bin gets exactly $k$ balls? And is this ...
2
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3answers
112 views

Constrained combinations of balls in jars

Not sure how to solve the following problem. Imagine we have balls of $n$ different colors. There are $m$ balls of each color, so in total we have $nm$ number of balls. The question is how many ...
0
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3answers
49 views

Count all events for k distinguishable items in n distinguishable bins

I want to count all the possible cases that I can distribute k distinguishable items in n distinguishable bins, for example for 3 items and 2 bins I have the next n^k the possible cases are: I ...
1
vote
1answer
44 views

What is the probability of picking $4$ balls out from the two bins?

Assume there are $2$ bins (bin $a$ and bin $b$ ), and each bin contains $n$ indistinguishable balls with $n>10$. The probability of picking $k$ ball from each bin is $p_k$, ($k\ge 0$). Then what is ...
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1answer
77 views

Combinatorics with balls and bins with constraint

I have $90$ identical balls to distribute among $64$ distinguishable bins. Each must get at least 1 ball, after that the distribution over the remaining $26$ does not matter. I know I first have to ...
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2answers
63 views

probability that given n balls and 60 urns no urn has more than 1 ball

A carnival game is set up so that a ball put into play has an equal chance of landing in any of 60 different slots. The operator of the game allows you to choose any number of balls and put them all ...
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0answers
61 views

Balls and bins question with nlogn balls and n bins.

The question: Suppose we randomly drop nlogn balls into n bins. Give an upper bound on the expectation of the maximum number of balls in any bin. How would this be done? I believe the answer is well ...
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2answers
46 views

Inclusion/Exclusion Principle for indistinguishable balls into two different types of boxes

The problem asks me how to distribute n balls into h boxes where each must get at least 1 n and t boxes where each can be left empty What I have, and where I seem to have stalled is how to ...
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2answers
125 views

Balls and bins problem - expected number of balls needed to throw [closed]

Suppose we have n boxes and we start randomly and independently throwing balls into the boxes. (a) For a given box, what is the expected number of balls we need to throw before one of the balls lands ...
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0answers
59 views

How to estimate pigeonhole principle?

I was thinking about this after my professor mentioned the pigeonhole principle in class. Let's say we have $N$ items and $M$ containers. Here we assume $N > M$. We will randomly place each of the ...
0
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1answer
178 views

Ball and bins & Coupon collector problem

If you know a coupon collector problem, you will know what I am talking about. But if you are not familiar with I will try to explain what is the coupon collector problem. I have $n$ bins. I throw ...
2
votes
1answer
131 views

Bins and balls model - filling first bins [close]

We have $n$ bins and $m$ balls. I want to compute the probability that in the first $k$ bins, $q$ of them will be non-empty. I can throw $m$ balls into $n$ bins in $n^m$ ways. Using Stirling ...
0
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2answers
62 views

Probability and conditional?

A bag contains four red balls and four white balls. A ball is selected at random, removed and replaced by a ball of the opposite colour. A second ball is then selected at random. Calculate the ...
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2answers
109 views

How many ways to $22$ balls in $5$ boxes problem

How many ways are there to put $22$ identical balls into $5$ boxes, with each box has at least $2$ balls? The answer is $16$ choose $4$, i.e., $_{16}C_4$, but can someone explain it to me?
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2answers
137 views

Probability that $\frac{n}{2}$ bins are empty [close]

A Bloom filter of length $n$ was built. I have only the first $\frac{n}{2}$ bits of this filter. How will the false positive probability change? For the whole Bloom filter, the false positive ...
0
votes
1answer
58 views

probability of specific number of balls in an urn

I have $n$ balls, I distribute them between 2 urns. Each ball goes to urn 1 with probability $p$ and urn 2 with probability $1-p$. What is the probability that there are $l$ balls in urn 1 when all ...
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0answers
113 views

The probability of two balls going into the same bin

Suppose I have n bins and n balls with label on them. What would be the probability that I throw the i th ball into the same bin as the j th ball? Is it just 1/n because there is a probability of 1/n ...
0
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1answer
74 views

Balls and bins with 2 balls and 2 bins

I'm trying to understand a proof in a paper I'm reading. It relies on a balls and bins problem. Here is what I'm trying to figure out: We want the maximum number of balls in a bin. We have 2 balls ...
0
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1answer
504 views

Number of ways to distribute identical balls into identical bins

I have a question which I expected to be quite famous and common, yet I haven't found much... How many ways are there to distribute $k$ identical balls into $n$ identical bins? For example $(k, ...
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1answer
232 views

Expectation of throwing $n$ balls into $n$ bins

Suppose we throw $n$ indistinguishable balls in $n$ bins at random. The throws are independent. What is the expected number of empty bins? What is the expected number of bins with one ball. Using ...
2
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0answers
66 views

Maximum bin load for $\alpha n$ balls into $n$ bins

In a paper I am reading the author writes: A standard result concerning balls and bins shows that if we throw at least $\alpha n$ balls into at most $n$ bins, then the maximum bin load is ...
0
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1answer
511 views

Balls and bins probability problem

$k = \sqrt n$ balls are thrown into $n$ bins. The bins are standing in a row and numbered from 1 to $n$. What is the probability that there are no two balls in the same bin or in adjacent bins??? In ...
0
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2answers
147 views

A bin and balls problem

Throw 7 balls into 7 bins. Given there are exactly 2 empty boxes, find the probability that 1 bin contains 3 balls, and thus the other 4 bins contain 1 ball each. I know I could use a multinomial ...
2
votes
1answer
409 views

Distribution of distinct balls in identical boxes

how can I derive a formula for the number of distributions of $n$ different balls in $k$ identical boxes. Where $\mathbf{empty\ box}$ is allowed. I know this is equivalent to finding the number of ...
2
votes
1answer
117 views

Bounds for max load with $n$ balls in $n$ bins?

If you throw $n$ balls into $n$ bins uniformly and independently at random, let $X$ be the number of balls in the bin with the largest number of balls in it. Is there a nice elementary way to ...
4
votes
3answers
674 views

Expected max load with $n$ balls in $n$ bins?

If you throw $n$ balls into $n$ bins uniformly and independently at random, let $X$ be the number of balls in the bin with the largest number of balls in it. Is there an elementary way to ...
0
votes
1answer
66 views

Expected number in b bins with max k balls each, balls in bins question

Suppose we have b bins and each bin could have max k balls. We have sufficient balls and what is the expected number if we need to full all bins? I know the answer when k = 1, the expected number is b ...
2
votes
1answer
311 views

Balls and bins conditioned on the number of non-empty bins

The expected number of occupied bins in the standard balls and bins problem (with $m$ balls into $N$ bins) is $N\left( {1 - {{\left( {1 - \frac{1}{N}} \right)}^m}} \right)$. How do I compute this ...
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1answer
141 views

$n$ balls into $k$ baskets, $n \geq k$, no empty baskets

I have $n$ balls and throw them into $k$ baskets. None of $k$ baskets should be empty. Which means each basket has at least one ball. Balls and baskets are not distinguishable. What is the number ...
2
votes
1answer
202 views

Average number of bins occupied when throwing $n$ balls into $N$ bins

There are $n$ balls and $N$ bins. At each time, a ball is thrown in one bin of $N$ bins at random. This repeats n times. So that in total $n$ balls are thrown into bins. The question is, on average, ...
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2answers
105 views

Urn problem: replacing white balls with black once selected

I'm trying to find the probability of an outcome where, using the traditional example, white balls are replaced by black balls once selected. Initially I have $n$ white balls and $\mu$ samples. I ...
2
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0answers
126 views

Modified balls and bins

Assume that we have $n$ balls and $k$ bins. The $n$ balls are are divided into $M$ sets, $\left\{ {{m_i}} \right\}_{i = 1}^M$, where $\sum\limits_{i = 1}^M {\left| {{m_i}} \right|} = n$, $|m_i| \le ...
0
votes
2answers
170 views

Probability in ball coloring

You have exactly $n^2$ balls each one of which can be colored in one of $n^2$ ways. That is total colors is $n^2$ but I am not saying all the $n^{2}$ balls are distinctly colored. However assume each ...
0
votes
1answer
125 views

balls and bins - the probability of reaching the exact expected value

Suppose I throw $n$ balls to $m$ bins (randomly and independently). The expected value of number of balls in each bin is $n/m$. How can I tell (or bound) the probability that every bin has exactly ...
2
votes
2answers
238 views

Probability of number of unique numbers in $37$ Roulette Wheel spins.

I was wondering if someone could help me answer the following question: Calculate the probability that in $37 $ consecutive spins of a roulette wheel (using an European Roulette wheel) you will have ...
2
votes
1answer
104 views

In how many ways I can put $2$ red balls and $3$ green balls in $5$ boxes?

I have $5(N)$ boxes and some balls. Here's the description: Red Balls $= 2 (k1 = 2)$ Green Balls $= 3 (k2 = 3)$ ...
0
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2answers
112 views

Yet another balls and boxes problem; minimum number of throws so that we have no empty boxes.

I managed to figure out how many empty boxes will be left given n amount of throws, just having a hard time figuring out the minimum number of throws necessary so that we have no empty boxes. Would it ...
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1answer
161 views

How many expected people needed until 3 share a birthday?

I asked a somewhat related question recently and then became interested in this one: how many people are required, on average, until 3 share a birthday? More generally, if we have $M$ bins, what is ...
2
votes
0answers
91 views

$m$ balls into $n$ already occupied urns

Assume that there are n urns, $k\in\{1,...,n-1\}$ of which have already at least one ball. $m$ new balls will be thrown into urns. Each of the $m$ balls is thrown randomly and uniformly into $n$ urns. ...
4
votes
1answer
238 views

$m$ balls into $n$ urns

Assume that there are $m$ balls and $n$ urns with $m>n$. Each ball is thrown randomly and uniformly into urns. That is, each ball goes into each urn with probability 1/n. What is the probability ...
5
votes
3answers
255 views

Second pair of matching birthdays

The "birthday problem" is well-known and well-studied. There are many versions of it and many questions one might ask. For example, "how many people do we need in a room to obtain at least a 50% ...
9
votes
3answers
404 views

Probability $k$ bins are non-empty

The following problem arises in the analysis of Bloom filters. Consider $m$ bins and $N=nk$ balls placed uniformly and independently at random into the bins. A query chooses $k$ bins uniformly and ...
7
votes
2answers
291 views

Find: The expected number of urns that are empty

A total of $n$ balls, numbered $1$ through $n$, are put into $n$ urns, also numbered $1$ through $n$ in such a way that ball $i$ is equally likely to go into any of the urns $1, 2, . . . , i$. Find ...