# Tagged Questions

4answers
71 views

### $n$ balls into $n+1$ urns (with one special urn)

Assume that there are $n$ balls numbered from $1,2,\ldots,n$ and $n+1$ urns, numbered as $0,1,\ldots,n$ Throw each ball randomly into one of $n$ urns: urn 1, ...
0answers
109 views

### Balls and Bins problem with constraint

Assume we have $B$ black balls and $R$ red balls, where $R+B$ is a multiple of 4. We want to distribute the balls in $\frac{R+B}{4}$ bins such that each bin has at least 1 red ball and at least 1 ...
2answers
67 views

### How many boxes will be empty?

$150$ balls randomly put into $100$ boxes, each ball could be put into any of these 100 boxes with same probability, after that, on average, how many boxes will be empty? No calculator. Choose one of ...
2answers
67 views

### Probability of red ball i before any black ball

Assume we have $r$ red balls and $b$ black balls in a box and we remove one ball at a time without replacement. Red balls are labeled from $1$ to $r$. We want to calculate the probability a particular ...
0answers
35 views

### Poisson Distribution?

There exists 1000 boxes. These boxes are randomly filled with balls. How many balls are required in order that only 1 in 100 boxes are left empty? This sounds like a Poisson distribution problem to ...
1answer
148 views

### Repeatedly Toss Balls into Bins

$n$ balls are randomly tossed into $m$ bins, each bin can hold $k$ balls. If a ball is tossed into a full bin (already has $k$ balls in it), it can be tossed repeatedly and randomly into the $m$ bins ...
1answer
77 views

### what is the expected number of days in year in which exactly k people in a group of n people have been born?

there is a group of n people and we must find the average number of days that in each of them exactly k people are born (k and n are given). This question assumes that a year has 365 days, and each ...
1answer
56 views

### $k$ balls into $n$ bins — Number of occupied bins

Suppose we throw $k$ balls into $n$ bins. Assume that $\log^2n\le k\le n$. Is there a high probability bound (preferably exponential) on the number of occupied (i.e., non-empty) bins? Something ...
0answers
48 views

### balls and bins: the first time when max-loaded is less than twice min-loaded

We have $n$ bins, in each step we throw a ball in a bin chosen uniformly and independently from the $n$ bins we have. We repeat the process $k$ times. Let $B_k$ be the number of balls in ...
1answer
109 views

### Bins in balls where bin size grows exponentially

I have $k$ bins. The first bin can fit $1$ ball. Each subsequent bin can fit two times more balls than the previous one. In other words, the $i$th bin can fit $2^i$ balls. We randomly assign \$U = ...
1answer
105 views

### Amount of distinct numbers in a sequence of $k$ random numbers in range $[1,\ldots,n]$

Let $D$ be the amount of distinct numbers in a sequence of $k$ random numbers in range $[1,\ldots,n]$ (n>k). I want to show that: $D=\Omega(k)$ with exponential high probability. I'm interested in the ...
2answers
140 views

### Throwing $k$ balls into $n$ bins.

I have the following question: Throwing $k$ balls into $n$ bins. What is the probability that exactly $z$ bins are not empty? I thought about something like: \Pr(z)=\frac{n! z^{k-z}}{n^k ...
1answer
114 views

### Maximum load of a bin in the $n$ balls with weights into $m$ bins problem

$n$ balls, each with a weight $p_i$, are thrown into $m$ bins. Each bin is chosen with uniform probability. Prove or disprove that the expected value of the maximum load among the loads of bins is ...
1answer
145 views

### Probability - Balls and Buckets; variance question

I've been working on this problem for a while and its giving me no end of trouble! The question is this: Suppose we have 2k buckets, numbered 1 through 2k. We throw x black balls and y white balls, at ...
5answers
193 views

### Probability that all bins contain strictly more than one ball?

Here's the problem I'm working on: Given that I'm distributing $N$ balls into $K$ bins, what is the probability that all bins contain at least two (strictly more than 1) balls? This seems like a very ...
1answer
150 views

### Yet another balls and bins problem

If $p_n$ denote the probability that when $n$ balls are randomly put in $n$ bins then there is at least one bin with exactly one ball. Is there a simple (involving only little computation) reason for ...
1answer
100 views

### Expected value for a function concerning a balls and bins problem

I'm optimizing a hash function mapping $M$ items into $N$ bins and I need a criterion for evaluating the quality of the mapping. Denoting the number of items put into bin $i$ by $x_i$, an ideal ...
1answer
750 views

### If n balls are thrown into k bins, what is the probability that every bin gets at least one ball?

If $n$ balls are thrown into $k$ bins (uniformly at random and independently), what is the probability that every bin gets at least one ball? i.e. If we write $X$ for the number of empty bins, what ...
1answer
214 views

### Hyper Birthday Paradox?

There are $N$ buckets. Each second we add one new ball to a random bucket - so at $t=k$, there are a total of $k$ balls collectively in the buckets. At $t=1$, we expect that at least one bucket ...
3answers
2k views

### How can I solve bins-and-balls problems?

Below is the problem that I wanted to solve When there are $m$ balls and $n$ bins, balls are thrown into bins where each ball is thrown into a bin uniformly at random. What is the expected number ...
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