Questions tagged [combinatorics]

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

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19 views

Mixed order and unordered counting

If you care about ordered or unordered counts you can use nCr or nPr. But If you want a few in a certain order and a few unordered is there a general approach? For example imagine you have 6 red ...
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12 views

Markov Chain Monte Carlo for Decryption Purposes

I've been reading the "Markov Chain Monte Carlov Revolution" paper by Diaconis and was intrigued by his application to decryption. I've done some further reading and am now trying to work through ...
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1answer
23 views

Number of Model Combinations

I have a model selection combinatorics problem. My covariate pool consists of six covariates: A1, A2, A3, B, C and D. However, covariates A2 and/or A3 can only be included if A1 is also included. ...
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36 views

probability/counting problem

This is an even question from the textbook so unfortunately there's no solution. Suppose $\DeclareMathOperator{\P}{\mathbb{P}} S = \{1,2,3,4,5,6,7,8\}$ with $\P({s}) = 1/8$. Part C) asks How many ...
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24 views

Formulating a mixed integer programming problem with a binary maximization term

Real-world description of the problem: In the US, funding for school lunch programs is given to local school districts by the Federal government. If >=40% of students in a school or group of schools ...
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1answer
18 views

Probability of group wins in a lottery: Same versus different first digit in a lottery number drawn digit by digit

I want to calculate the probability to win in a certain lottery. These are the rules: In the lottery, participants can participate individually or as a group. Groups can have up to 200 members. Each ...
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1answer
24 views

Finding the expectation: Draws to get 2 gold coins

Omar has saved 11 precious coins, 7 of which made of gold, in a jar. He draws the coins one by one from the jar, to find the gold coins. If $X$ is the number of coins drawn until he has found the ...
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1answer
25 views

Drawing six cards from a deck of cards

Six cards are drawn from a well-shuffled deck. You are told that at least an ace (A) is among the cards. What is the probability that there is at least a king (K) or at least a queen (Q)? (A deck ...
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1answer
27 views

How to solve this combinatorial problem about selecting letters from different groups?

The English alphabet consists of 26 letters, 6 of which (A, E, I, O, U, Y) are vowels and the remaining 20 are consonants. If each letter has the same probability of being chosen, what is the ...
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1answer
136 views

Probability and Counting

Suppose You are ordering two pizzas.A pizza can be small,medium,large or extra large,with any combination of 8 possible toppings( getting no toppings is allowed, as is getting all 8) How many ...
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2answers
64 views

How to choose the best combination of covariates in Cox multiple regression?

I am performing a multivariate Cox regression analysis, and would like to find what combination of those covariates best predict my outcome. Say I have a list of candidate genes whose expressions ...
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1answer
34 views

Item sampling techniques to reduce processing time for questionnaire

I have a well-crafted draft of a questionnaire, containing 240 items. The average processing time is 20 secs per item, so the entire questionnaire would take around 1 hour 20 min to complete. I want ...
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20 views

All different size groups to create m groups from n samples

Suppose we have $n$ samples that we want to divide in $m$ groups. How can we get the total number of groups and their sizes (if possible)? Examples: For $n = 6$ and $m = 3$ we would have $3$ groups as ...
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17 views

Counting the number of tuples $(i,i',k,k')$ satisfying some conditions

I'm struggling to find the cardinality of a set that will be presented below, as well as an upper bound for it. It is an engaging problem and perhaps not trivial. I will give now the baseline of the ...
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2answers
112 views

What is the probability of having a pair of doubles when throwing dice?

What is the probability of having a double (two dice will show the same number) when we throw 5 dice (all dice have 6 sides)? My calculations are: ...
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9 views

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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14 views

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
18 views

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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3 views

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
21 views

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
78 views

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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19 views

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
127 views

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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2answers
48 views

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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22 views

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
37 views

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
16 views

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
12 views

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
465 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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31 views

Probability of having a match in an existing database [on hold]

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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58 views

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
27 views

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
56 views

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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28 views

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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28 views

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
36 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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26 views

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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30 views

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
106 views

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
102 views

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
32 views

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
41 views

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
74 views

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
130 views

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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145 views

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
27 views

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
20 views

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
23 views

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....