0
votes
1answer
18 views

Distribution combinations

How many ways can 25 identical pencils be distributed between two people?.Each all pencils must be shared out. A) Each person must have at least 5 pencils? B) Each person must have at least 7 ...
1
vote
0answers
44 views

The probability that exactly / at-least $k$ numbers are in the correct position [duplicate]

Given a sequence of $[1,\dots,n]$ in random order: Let $P_k$ be the probability that exactly $k$ numbers are in the correct position Let $Q_k$ be the probability that at least $k$ numbers are in the ...
3
votes
1answer
64 views

What is the joint probability distribution of number of balls after $n$ draws?

The following problem came into my mind when I am studying the Polya Urn Model. At the beginning, from a bin containing $c_1$ balls labeled $1$, $c_2$ balls labeled $2$, … , $c_m$ balls labeled $m$, ...
0
votes
1answer
18 views

On a real line R points a,b are randomly selected such that -2<=a<=2 and 0<=b<=3. Find the probability that | a - b | > 1

Let's say that C is the set where |a-b|>1 So I suppose you could say plot it as coordinates where the x-axis (labelled a) is from [-2,2] and the y-axis (labelled b) is from [0,3]. Now |a-b| must be ...
2
votes
1answer
59 views

Generating function for picking j balls without replacement from an urn

In an urn, each balls is labeled with one of $\{0,1,2,...,k\}$. For each $i\in{0,1,2,...,k}$, there are exactly $n_i$ balls labeled $i$. Let $f(x)=\sum\limits_{i=0}^k n_ix^i$. Let ...
1
vote
2answers
53 views
5
votes
2answers
2k views

A fair coin is flipped 2k times. What is the probability that it comes up tails more often than it comes up heads? [duplicate]

I'm studying for a probability exam and came across this question. I watched the video solution to it but I don't really understand it. I was hoping someone could explain this problem to me. Are there ...
0
votes
1answer
84 views

Help needed to derive combinatorics formula.

I am having troubles understanding a combinatorics formula. I would appreciate any ideas or hints, leading to an explanation how this formula might be derived. I came across the formula reading a book ...
0
votes
0answers
21 views

Pairwise independent subsets of fixed size

For a given set $\Omega$ of even size $n$, does there exists, in general, a collection of subsets $F \subset 2^\Omega$ such that $\forall A \in F. |A| = \frac{n}{2}$, $\forall \omega \in \Omega. ...
0
votes
0answers
130 views

Formula when drawing from multiple urns (Probability)

I'm trying to work out a general solution for a probability problem, but I can't seem to figure out how to go about with it. I manage to calculate the individual probabilities on a per problem basis, ...
0
votes
0answers
57 views

Marginal and conditional probability table without joint probability table

I've a Bayesian network, with discrete node values: for every node I've the conditional probability table $p(A|B)$, where $A$ is the node itself and $B$ is the set of the parents nodes. Now I would ...
0
votes
0answers
16 views

combining continuous distributions

If we have several continuous distributions, for example ten Beta distributions, how we can combine them by the linear and log-linear opinion pool methods? I know how to combine discrete ...
1
vote
2answers
29 views

Division of Objects into Different Sized Boxes

Suppose you have a set of N distinguishable boxes with lengths $l_1$,$l_2$...$l_N$. Suppose you try to divide x distinguishable objects among them, such that the probability of any object landing in ...
1
vote
1answer
46 views

determining the size of a test bank given acceptable number of repeats

I have a question for a challenge that I'm trying to create - having some trouble quantifying the size of the challenge's test bank. 20 people are taking a challenge of 9 questions the test bank (n) ...
10
votes
3answers
690 views

A very challenging probability question

In a certain 2-player game, the winner is determined by rolling a single 6-sided die in turn, until a 6 is shown, at which point the game ends immediately. Now, suppose that k dice are now rolled ...
1
vote
1answer
37 views

An extended “birthday sharing” problem; sharing multiple properties

== THE SETUP == The table below shows the probability that Person N will draw a specific color ball out of a bag. Each person is given one opportunity to draw each color ball. For example, person 1 ...
2
votes
1answer
49 views

Probability Question I can't get around.

This is the question from my assignment, which I can't get around. Suppose that a water distribution system is composed of a number of independent pipes. At temperatures below 0 deg C, the pipes ...
1
vote
4answers
73 views

issues with probability

a man has $4$ children, given that atleast one of whom is a girl.Find the probability that he has $3$ girls and $1$ boy. MY TRY : probability of girl=$1/4$ and probability of boy=$3/4$ (my doubt is ...
0
votes
1answer
44 views

Lottery Ball problem

There are two sets of numbered balls. One set consits of white balls numbered $1-10$ the other is blue balls numbered $1-20$. To play you select two white balls and two blue balls. What is the ...
1
vote
1answer
62 views

Drawing without replacement, a special case

I have the following problem: you pick a set $x$ containing $|x|$ elements from a bag, containing $k$ marbles of $m$ possible types. Once a marble of a certain color is drawn, however, all other ...
0
votes
3answers
39 views

Simple combinatorics help including boxes and objects

How many ways are there to distribute $k$ balls into $n$ distinct boxes ($k < n$) with at most one ball in any box if (a) The balls are distinct? (b) The balls are identical? My ...
2
votes
0answers
68 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
votes
0answers
143 views

factorial moments of hypergeometric distribution

Factorial moment of positive order : $$\mu_k=\mathbb E[X(X-1)\ldots(X-k+1)]$$ $$=\sum_{m=0}^{n}m(m-1)\ldots(m-k+1)\frac{\binom{a}{m}\binom{b}{n-m}}{\binom{a+b}{n}}$$ ...
1
vote
2answers
1k views

Probability of sampling with and without replacement

In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding ...
2
votes
3answers
407 views

Random number generator with discrete probability distribution

Is there a general algorithm for implementing a PRNG with a probability distribution?
1
vote
1answer
422 views

Negative binomial distribution - deriving of the p.m.f. combinatorially

Let $X$ be the number of trials preceding the $k$th success in a sequence of independent Bernoulli trials each with probability of success $p$. Then $X$ has a negative binomial distribution with ...
1
vote
1answer
63 views

Calculating the probabilities of different lengths of repetitions of numbers of length 4

I'm trying to calculate the probabilities of different lengths of repetitions of X length number however I know I'm doing it incorrectly since when I add all the probabilities together they don't ...
1
vote
0answers
66 views

Choosing at random with Replacement. pmf? E(x)?

There are $30$ balls in a box. $6$ of them are red, $10$ are white and $14$ are blue. $10$ balls are chosen at random with replacement. Let $X$ be the the number of red balls in the sample. 1) ...
1
vote
1answer
425 views

Joint density of order statistics $f_{X_{(1)}X_{(n)}}(x,y)$ with combinatorics

Problem I need to find $f_{X_{(1)}X_{(n)}}(x,y)$ for the uniform distribution. $X_{(k)}$ denotes the $k^{\text{th}}$ smallest from an n-sample. I already know that the answer is ...
2
votes
2answers
67 views

Calculating an “at least” probability without summation?

I know One can calculate the probability of getting at least $k$ successes in $n$ tries by summation: $$\sum_{i=k}^{n} {n \choose i}p^i(1-p)^{n-i}$$ However, is there a known way to calculate such ...
0
votes
0answers
156 views

Distribution of Levenshtein distances for partially sorted lists

I have a partially sorted list of distinct items and want to know the probability of this occurring by happenstance rather than intent. The Levenshtein distance is a good metric for the problem ...
4
votes
0answers
146 views

Using Bernoulli distribution approximate the $q$-th moment

Let $x$ be vector in $R^n$. Let $\pi(⋅)$ be a permutation on the set $\{1,\ldots,n\}$ with a uniform distribution. Let $|m|\leq n, m \in Z$. Using Bernoulli (or maybe some other) distribution ...
1
vote
0answers
97 views

Probabilistic results on the elementary symmetric polynomials

The elementary symmetric polynomials of degree $k$ in $N$ variables are defined as $$e_k(x_1, \ldots, x_N) = \sum_{(i_1,\ldots,i_N) \in I_k^N}{x_1^{i_1}\ldots x_N^{i_N}}, \quad 0 \le k \le N$$ with ...
0
votes
2answers
66 views

Combinatorics problem: probability density function for picking up element experiment

Suppose to have $n$ elements of $2$ different types. Let $n_1$ and $n_2$ be the numbers of elements of each type respectively (with $n=n_1+n_2$). I have to pick $k$ elements from this set. Every ...
0
votes
1answer
153 views

Is Lottery probability really the same for all combos?

http://justwebware.com/uklotto/uklotto.html Test run quickpick Test run 1,2,3,4,5,6 Test run (single digit,teens,twenties,twenties,thirties,forties) 1000 times or more each cycle for as many ...
5
votes
3answers
563 views

Compute probability of a particular ordering of normal random variables

There are $m$ normally distributed, independent random variables $N_1, \ldots, N_m$ with distinct means $\mu_1, \ldots \mu_m$ and standard deviations $\sigma_1, \ldots, \sigma_m$. Then, we get a ...
0
votes
0answers
74 views

Distribution of binary digits in moduli

Considering the (infinite) set of all positive integers that are a product of $2$ primes only, represented in binary $100...01$. Question: is the distribution of the proportion of $0,1$ digits ...
1
vote
1answer
409 views

Dependent Bernoulli trials

The probability of a sequence of n independent Bernoulli trials can be easily expressed as $$p(x_1,...,x_n|p_1,...,p_n)=\prod_{i=1}^np_i^{x_i}(1-p_i)^{1-x_i}$$ but what if the trials are not ...
0
votes
3answers
378 views

How many ways are there to distribute 2 indistinguishable white and 4 indistinguishable black balls into 4 indistinguishable boxes?

How many ways are there to distribute 2 indistinguishable white and 4 indistinguishable black balls into 4 indistinguishable boxes? How can we solve this?
4
votes
3answers
295 views

What is Cumulative Distribution Function of this random variable?

Suppose that we have $n$ independent random variables, $x_1,\ldots,x_n$ such that each $x_i$ takes value $a_i$ with success probability $p_i$ and value $0$ with failure probability $1-p_i$ ,i.e., ...
2
votes
1answer
84 views

probability convolution problem

Suppose $X,Y$ are uniformly distributed independent random variable on $\{1,...,N\}$ , compute the density of $X+Y$. So the density of $X$ or $Y$ is $f_X (x) = \frac{1}{N}$ (so if we sum the terms, ...
9
votes
4answers
2k views

Expected Value of a Binomial distribution?

If $\mathrm P(X=k)=\binom nkp^k(1-p)^{n-k}$ for a binomial distribution, then from the definition of the expected value $$\mathrm E(X) = \sum^n_{k=0}k\mathrm P(X=k)=\sum^n_{k=0}k\binom ...
4
votes
1answer
137 views

What is the probability distribution of a single genome base pair

in the genome we have 4 nucleotides (A,T,C,G). Now given a nucleotide sequence like AGT CG TA CG ATCT CG , we can count the number of "CG" pairs. That's 3 in this case. (we count all the pairs so, ...
1
vote
1answer
53 views

Noise sensitivity of Boolean functions

Is there any Boolean function from $\{-1,1\}^n$ to $\{-1,1\}$ such that whose noise sensitivity is greater than delta, where Delta is the probability of each bit is flipped in n-tupple.
4
votes
2answers
232 views

Joint density of the smallest and largest random variables among finite independent random variables with common density

I am trying to show the following result. Let $X_1, \ldots,X_n$ be independent random variables with the common density $f$ and distribution function $F$. If $X$ is the smallest and $Y$ the largest ...
5
votes
1answer
420 views

Multivariate Hypergeometric Distribution/Urn Problem

I am having a difficulty with the following multivariate hypergeometric distribution problem. The setting is as usual, an urn contains a total of $M$ balls of $K$ unique colors, with $N_1$ balls of ...
6
votes
1answer
325 views

How to prove this combinatorial identity?

I am wondering how to prove the following identity: $$\sum_{i=0}^{n-r} \frac{2^i (r+i) \binom{n-r}{i}}{(i+1) \binom{2n-r}{i+1}}=1?$$ It seems this might be related to the hypergeometric distribution, ...
8
votes
3answers
994 views

“Go-first” dice for $N$ players

I'm interested in sets of dice that can be used to determine who "goes first" (hence the name) in an $N$-player game; more generally, I want to determine a complete ordering of the players with a ...
3
votes
2answers
399 views

Generating function for Banach's matchbox problem

Here's the description for Banach's matchbox problem from Concrete Mathematics EXERCISE 8.46 (edited) Stefan Banach used to carry two boxes of matches, one containing $m$ matches and the other one ...
1
vote
1answer
153 views

How long does it take to complete a sticker album?

We are collecing stickers in chocolate bars and whenever we open a bar we get a random new sticker. There are many different stickers and we try to collect them all. We open the first bar and get a ...