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Questions tagged [combinatorics]

Counting or enumerating elements in a set or other finite discrete structure.

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Combinatorics of distinquishable sequences of events

I have found a problem that I can only solve by guessing, but without any systematic solution: 3 distinguishable balls (red, blue, green) 5 distinguishable sites (A, B, C & X, Y) X, Y can ...
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Estimating weights in the assignment problem

How would you learn a function with the emphasis on feature interactions? I have the standard assignment problem: $$ \max_{x_{ij}} \sum_{(i, j)} w_{ij} x_{ij}, $$ where $w_{ij}$ is the weight of ...
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1answer
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Find the probability that two samples of F unique items taken from a population of N unique items has at least one common item

What is the probability that two samples of F unique items taken from a population of N unique items has at least one common item?
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Calculating # of possible binary strings given constraint

Consider a random binary string where each bit can be set to 1 with probability $p$. Let $Z[x,y]$ denote the number of arrangements of a binary string of length $x$ and the $x$-th bit is set to 1. ...
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1answer
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Creating mean variables using all possible combinations of the original variables in r [closed]

I need to create a function that calculate the mean of all possible combinations of a given number of variables, and store the means values as separate variable and the number of variables whose means ...
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2answers
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sample all pairs without repeats

Assuming I have a very large number of $K$ colored balls and we know the fraction of each color. If we randomly sample all pairs so that all pairs have two balls of different colors then what is the ...
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All combinations choosing 3 items each from 4 unique bags

Trying to do something like expand.grid in R, but as if expand.grid created all combinations of sets of 3 from each factor/group. (E.G.)... ...
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3answers
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How to set up an event with probability of one seventh from coin tossing?

I have been recently asked in an interview a question that: How to set up an event with probability of one seventh from coin tossing? I couldn't find the correct answer and still don't know the ...
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combinatorics problem: two ways to solve same problem?

Currently i'm struggling alot with combinatorics. The problem: There are four siblings. What is the probability that at least two of them are born in the same month? This question asks to get two ...
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Probability of combination of letters appearing on multiple custom dice

Question: Say I have a set of 24 dice that have different letters on them, what is the chance that in any roll a certain combination of letters appears at least once? Combination I want: {a, a, b, c, ...
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1answer
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Modified coupon collector: killing traitors problem

You have $n$ people, $m$ of whom are traitors. You keep killing them randomly until all the traitors are gone (which you presumably have some way to verify). On average, how many people $X$ will you ...
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1answer
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Why is the formula for permutations of size one n!

I am looking at this formula I want to first make sure that I understand the principle. This is saying that, if I wanted to find out all the ways I could order the letter of the alphabet into a 26-...
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1answer
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Does the multinomial coefficient create combinations or permutations?

I am looking at this formula I just want to make sure I get it. For example, if I want to split a fifty-two card deck into all the possible combinations of four, thirteen card hands, then I get: 52!...
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1answer
235 views

Probability of normal random variable yielding highest value among other normal variables

Let's say I have a competition with $N$ participants. Each participant yields a score that is normality distributed with unique means and unique variations. Each participant gets to post one score (we ...
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Probability of having a match in an existing database

Imagine you are an airport authority in CountryLand and you have two passenger types: human passengers and animals. There is no way for you to distinguish between human passengers and animals except ...
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Probability to be present in every subgroup

Assuming you randomly split a set of n elements in $y$ equally sized parts(each has $x$ elements). Now you have a subset of m elements in the set of n elements ($m> (y/2)$). What is the probability ...
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1answer
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What is the probability that two samples from a source have no overlap?

I picked 29 results from a list of 429 results. I then picked a second group of 27 (with replacement) results from the same list of 429. There was no overlap between the two samples. What is the ...
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1answer
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How to calculate number of solid color images on the screen of random pixel generator?

Is there a statistical formula/function to calculate number of basic geometric shapes on the displays of random pixel generator? Let's say we have a basic random pixel generator which has 10*10 ...
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Birthday problem extension to unequal probabilities and multiple collisions

Let $p_1, ... ,p_k$ denote the probabilities of drawing bin $1, .. ,k$, where $\sum_{i = 1}^{k} p_i= 1$. My question is if we draw $n$ times, how can I show that the probability of no bin being drawn $...
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How many pairwise combinations is possible?

I have a dataset regarding a natural language problem that am tackling with machine learning. My goal here is the generate training samples for one of the classes. I do this by doing pairwise ...
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2answers
32 views

Checking my combinatorics

I am modelling a flash memory system where 0.3 requests are writes (and so take 100 cycles to complete) and 0.7 are read-only (and so take 50). The system can handle 4 requests at once (the ...
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1answer
35 views

Probability of drawing different items from a subset with replacement

I'm trying to generalize a game mechanic, to understand the probability that the game developers are giving us, but this went over my head. The mechanic is: draw 10 items from a bag of 33 items each ...
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finding the 'best' set

I came across this simple looking but puzzling question recently. There is a set of N tuples given [(a1,b1), ..., (aN, bN)], where a are real numbers and b are positive real numbers. We need to choose ...
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Approximating Binary Target Vectors using Machine-Learned Boolean Functions of Logical Statements

I was wondering if there exist any ML algorithms for finding the "best possible" Boolean function whose output will approximate a binary target vector. My data consist of roughly 100k observations (...
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1answer
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Inclusion/exclusion with identical sets that become distinct

This question is motivated by Blitzstein & Hwang Problem 1.49, but differs (I believe) in some crucial ways and so does not have the same answer. It is not (presently) an active problem on any ...
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Probability of recurrence of the same two numbers in two consecutive lottery drawings

Given lottery game 6 out of 49, I would like to know how high the probability of recurrence of the same two numbers (i.e. 6 and 12) in two consecutive lottery drawings is. In this Wikipedia article I ...
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1answer
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Probability you end up at the origin after taking $2n$ steps?

Starting at the origin on the line we take a step of unit to the left or to the right with probability $\frac12$. We do this repeatedly with independent steps. If we take $2n$ steps, what is the ...
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1answer
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Combinatorics with colored balls [closed]

Seven blue and four red balls are to be arranged in order. How many ways can this be done if (1) The blue balls are distinguishable (e.g. numbered) as are the red balls. (2) Blue balls are ...
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1answer
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Random Variables and Probability

So I encountered this problem while I was studying for exam. However, I cannot wrap my head around the solution that the answer key provided. The problem goes like this: Bob watches cars that pass ...
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1answer
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Have 40 people at 4 tables: how to switch tables so that the minimum number of people possible are sitting with someone at their original table

Just a few simple premises: 40 people, 10 people per table At some point you want every one to switch tables, and have a few people as possible sitting at a table with someone they sat with the ...
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2answers
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Calculating a MLE for the combinatorial probability distribution: $\sum_k p^k(1-p)^{N-j-k}\binom{N-j}{k} \cdot(1-p)^{i-k}p^{j-i+k}\binom{j}{i-k}$

I have a relatively complicated discrete probability distribution: $$\begin{aligned} P(i;j) &= \sum_k p^k(1-p)^{N-j-k}\binom{N-j}{k}\cdot(1-p)^{i-k}p^{j-i+k}\binom{j}{i-k} \\ &= \frac{p^j}{p^...
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Probability of sum of sequences of integers

Let K be a positive integer.Suppose that the integers 1,2,3,...,3k+1are written down in random order.What is the probability that at no time during this process, the sum of the integers that have been ...
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1answer
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Joint probability function of discrete variable (combinatorics)

There's a box with three types of objects: A, B, and C There are 6 of A, 8 of B, and 10 of C. At random, we remove four objects from the box I'm trying to find the joint probability $P(x, y)$ where $...
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1answer
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Finding number of seating arrangements

This is the question I got in Purdue University's Probability course available on YouTube: I don't understand they wrote 24 possible outcomes . Outcomes are just 5: 5 people in 5 seats. ...
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1answer
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How many ways are there to place 2 types of objects in n (that varies) spots?

First time posting, all help appreciated. I'm trying to figure out the number of ways to arrange items in spots. The complication is that the number of spots changes with the type of items put in them....
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1answer
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Bridge Hand Probabilities

Each of the 4 players in the game of bridge get dealt 13 cards. One player and his partner can see they hold 8 of the heart cards so they know that the 2 remaining hands they can't see hold the ...
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What is the name of this test for the probability of differences between lists and is it valid?

And are there any better methods? In a 1969 book about investing, the author Burton P. Fabricand describes a method for assessing the effectiveness of a stock picker. Ask the picker to provide a list ...
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1answer
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Different ways of counting the same combination

What’s wrong with this reasoning: How many different ways to pick a team of 3 from 4 people? ${4 \choose 3}$ Alternative way of counting is to choose a team of 2 from the 4 first (${4 \choose 2}$ ...
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Realisation of at least one event among N events

Two dice are thrown r times. I want to find the probability $p_r$ that each of the six combinations (1,1),...,(6,6)appears at least once. My Answer:-$p_r=6*(1-(\frac{35}{36})^r)-15*(1-(\frac{34}{36})^...
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3answers
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Coin Tossings probability

I want to find the probability that in ten tossings a coin falls heads at least five times in succession. Is there any formula to compute this probability? Answer provided is $\frac{7}{2^6}$
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Predicting a combination

Question Suppose we have a training set of families. Where each family is defined as such… Family: A list of integers. Each integer is the age of one of the family members. (e.g. with a 45 year ...
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Optimizing a sequence of events on a variable conditional on the event

I'm trying to understand how I would approach a problem aiming to optimize a sequence of events to maximize another variable. For example, say I start selling a t-shirt at multiple stores, where each ...
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1answer
122 views

Feller shoe-matching problem

This problem have been taken from the book' An Introduction to Probability Theory and Its Applications' by Williams Feller(1906-1970) Note:- Assume in each case that all possible arrangements have ...
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1answer
29 views

Randomly choose 50% of n things, but only 1% of specific k things implies what?

Summary: If you randomly choose 50% of n entities total, but it turns out you only chose 1% of k specific pre-chosen entities, does that imply n is likely to be much larger than k? I'm trying to use ...
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1answer
44 views

Probability of single entry originating from same individual

If I have a bag with n balls of k colors (equal number of each color), and I draw x balls (with replacement), what is the chance of getting two of the same color? I tried to simplify but not sure if ...
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1answer
38 views

Difference between permutations and combinations

Ok, I know this question has been asked a thousand times. I do understand that permutations mean that order matters where combinations not. I know there's more to it, like what we are counting and the ...
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Probability that 3 out of 4 sets have non-empty intersection

I have $4$ randomly sampled subsets from population set $S = \{1,2,...,100\}$. Each subset has size $24$. What is the probability that at least one element is common among $3$ out of $4$ subsets?
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Combinations or Counting general formula

Assume you have 5 numbered bins and you have 2 red balls, 2 blue balls and 2 green balls (You can assume that there are equal number of red, blue and green balls). I am trying to derive a general ...
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1answer
118 views

Does an approximation exist between the Cumulative Binomial Distribution and the probability of combinations?

Suppose that we have two disjoint subsets $A$ and $B$ of $Z$. Both $A$ and $B$ have a large number of elements each. We want to compute the probability of picking $k$ elements from $A$ e $B$, with ...
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1answer
109 views

Why do we use ${n\choose k}$ for a binomial distribution instead of ${n+k-1\choose k}$?

I am trying to get my head around this. In my understanding a binomial distribution uses replacement and ${n\choose k}$ precisely states that there's no repetition and that's not the case with a coin ...