Questions tagged [combinatorics]

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

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18 views

Use sum of p-values from significance tests to determine best permutation of table columns

I have a table in which there are columns which are entirely filled with numbers. Given another table, I want to figure out which permutation of the other table's columns produces an arrangement most ...
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12 views

How would you work out the total number of possible plays in a game of dominoes?

Looking at a standard double-six dominoes set that has 28 tiles 0:0 through 6:6 how would we calculate (or estimate) the number of possible moves in a single game? For this so far I've come up with: ...
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51 views

Amount of Possible Bootstrap Samples

I know that the total possible amount of distinct bootstrap resamples of size $n$ is $\binom{2n-1}{n-1}$. Is there any way to prove or justify this?
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Optimization Technique For N Dependent Bets

I was reading about Kelly betting, and found this article: https://www.pinnacle.com/en/betting-articles/Betting-Strategy/the-real-kelly-criterion/HZKJTFCB3KNYN9CJ In it, the author determines the ...
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1answer
31 views

Closed form of pairing probability

This situation happened to me at home today, and now it is bugging me. I didn't put all my clothes in the washing machine, and I retrieved 7 sockets, 2 pairs of socks and 3 unpaired ones. I wanted to ...
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Probability of x out of G objects sharing a bucket with the R randomly distributed other objects in n buckets? [closed]

If I have n buckets, and there are G green balls randomly put in those buckets, one at a time, and I put R red balls randomly, also one at a time, in those n buckets, what is the formula for the ...
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23 views

Drawing from 2 bags of different size and counting the number of outcomes

I have a combination problem that I can't get my head around. Similar problems were found here and here but are not quite the same of my problem. I have two sets of units and I want to find the ...
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43 views

Combinatory probability, group composed by different balls of different colors

Let's say you have a group of $M$ balls of different colors in a box. For example, 20 balls are red, 15 are blue, 10 are green, 5 are grey, 5 are yellow and 5 violet, for a total of $M=60$ balls. ...
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103 views

Combinatorial explanation for $2n$th moment of standard normal distribution?

The $2n^\text{th}$ moment of the standard normal distribution $X \sim \text{N}(0, 1)$ is given by: $$\mathbb{E}(X^{2n}) = \frac{(2n)!}{2^n n!},$$ which is also the number of ways to divide $2n$ ...
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23 views

Difficulty Understanding Application of the Multiplication Principle

While leafing through "Introduction to Probability" (Hwang, Blitzstein), I encountered the following problem. A standard deck of cards is shuffled well. Two cards are drawn randomly, one at ...
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1answer
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Probability and Combinatorics intuition behind matching birthday paradox

STATEMENT Suppose that a room contains $n$ people. What is the probability that at least two of them have a common birthday? Formula is $P(A^c ) = 365 \times 364 \times· · · \times (365 − n + 1) \...
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Is there value in assigning a dimension-specific weighting when calculating the main effect?

I've been looking into factorial designs and noticed an interesting property. Is this simply a mathematical trick or can we attribute some 'meaning' to it? Say we have a $2^k$ design where k = 3 here....
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41 views

Unexpected result from very simple statistics problem

I have stumbled across this problem on my mathematics book, the problem states: A person is distributing 3 packages in a city consisting of 3 houses, each package assigned to a single house. He has ...
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2answers
46 views

Question about Uniform Random Permutation

I'm reading Chapter 5 of the CLRS Algorithms book. Specifically, it discusses the hiring problem in which we want to assess the expected number of hires given that candidates arrive in random order ...
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Subset selection with reinforcement learning

I have a set of $N$ items of which a subset of arbitrary size can be chosen. I want a reinforcement learning (RL) agent to perform the subset selection and am unsure how to best design the action ...
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1answer
45 views

Which statistic is this 'overlap between sets/sections' - if any?

I’m sure someone in history already invented the following formula before. Does anyone know if so and what it’s called? Let’s say there is a set of number pairs, such that the second number in each ...
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1answer
29 views

number of ways to arrange n black balls and n white balls on a circle

Say you have N white balls and N black balls. How many distinguishable ways can I arrange them in a circle? I consider 2 arrangements to be distinguishable if no rotation or reflection can make them ...
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Survey of Deep Learning Linear Combinatorial Optimization

I have the task of developing an AI based combinatorial optimization algorithm involving a complex mix between TSP and Knapsack: A path must be found connecting millions of nodes and the cost ...
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1answer
41 views

Exact distribution of number of transitions in a binary vector of length $n$

I am trying to derive an analytic solution for the exact distribution of the number of transitions in a binary vector of length $n$ conditional on the observed number of $1$'s and $0$'s and under the ...
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1answer
45 views

Identification of special numbers based on recursive formula (combinatorics)

I am in the middle of a development and I stumbled on recursively defined numbers. They don't ring a bell for me, but maybe they were already studied. Anybody has a hint? $$ \kappa^{(k+1)}_{\delta} =...
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1answer
17 views

Probability of observing a number of special balls from a larger set

Suppose there is a collection of $n$ balls of which $m$ are special. What is the probability of drawing $k$ special balls, when $p$ balls are drawn? To give it a try I considered the following ...
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Proper length of random nonce for hash calculation (blockchain)?

I have a string s and I need to calculate a nonce such that when appending the nonce to the string, the generated hash starts with a given sequence. The hash has 256 bits, so the number of ...
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1answer
51 views

What are the most and least likely samples in a bootstrap?

This question is question 60 of chapter 1 of Introduction to Probability by Blitzstein and Hwang. Basically the question walks you through reasoning about the bootstrap sampling procedure. Given $(...
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Best practice for obtaining most suitable weighed combination of variables to predict data series (using R)

I'm already pondering for quite some days on a possible solution for, what I think, is an optimization problem. Despite the various attempts to find a suitable approach or guideline online, I am not ...
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What is statistical way of determining that questions are chosen at random from a test bank?

I'm using a website for professional certification that provides practice tests with questions generated from a question bank. The site claims that questions are chosen at random from a question bank ...
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How many ways are there of arranging the letters in the word [duplicate]

I am trying to understand my exercise but I still couldn't understand my teacher so I wonder if anyone please can help me? How many ways are there of arranging the letters in the word BANANA? The ...
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Misunderstanding Permutations with Repetition

The wikipedia entry on Permutations has this section on Permutations With Repetition: Ordered arrangements of the elements of a set S of length n where repetition is allowed are called n-tuples, ...
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1answer
126 views

Balancing out in an orthogonal design

A definition of orthogonality in the context of statistics is An experimental design is orthogonal if the effects of any factor balance out (sum to zero) across the effects of the other factors. ...
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32 views

How many combinations of students possible? [closed]

We have 1000 students and want to know what the maximum number of combinations there are of student groups where there are 3 students in each group and no student can ever be in a group with a student ...
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1answer
210 views

Limit distribution associated with counts (non-trivial combinatoric problem)

Update on 3/1/2020. All the material below and much more has been incorporated into a comprehensive article on this topic. The question below is discussed in that article, entitled "State-of-the-Art ...
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1answer
41 views

Generating n distinct combinations from a weighted distribution

If I'm not mistaken, this is an extension of this question: How to resample in R without repeating permutations? I have a vector x representing a non-uniform ...
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1answer
777 views

How many ways there are to add up 7 numbers to sum of 180?

Farmer has 7 pigs and 180 potatoes. How many combinations of distributing those 180 potatoes there are? Order does not matter. Each pig can get from zero to 180 potatoes, but all 7 pigs must get ...
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Probability of observing the same individual in randomly selected groups?

Suppose you observe $N$ individuals in $4$ years (the group of individuals is always the same). In each of those $4$ years, $Y$ individuals are selected are random (e.g., in each year, we select $10$ ...
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1answer
26 views

Interpret several classifications to one classification result

I want to interpret several data series X, that consist of like 4 to 20 values that can range from 1 to 5. Imagine these as a result of classifications, where each row is a different id, and each ...
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1answer
361 views

Balanced incomplete block design

In a BIBD, let $a$ denote the number of treatments; $b$ denote the number of blocks; $k$ denote the number of treatments exactly in each block; $\lambda$ is the number of times each pair of ...
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19 views

Calculating probabilities of completing a trading card/collector card set [duplicate]

I'm looking at creating a set of collector cards (think baseball cards). I'm worried about the statistics and would like some help if possible. For the sake of simplicity, let's say the basics are: ...
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25 views

Difference order distribution of extremal values of two uniform sequences

Fix an integer $m$. Pick two random integer sequences ($a_1$ to $a_n$ and $b_1$ to $b_{n'}$) uniformly independently from $[1,m]$. Denote $a(i)$ to be $i$th smallest value of $a$ and $b(j)$ to be $j$...
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1answer
46 views

Poker hand: probability of a straight or a straight flush

I have seen this and this, but my setup is a little different so I'm not sure how to proceed. Setup: We are using a single standard deck of 52 cards. Suppose I have been dealt two cards, and one of ...
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1answer
24 views

How do you work out the running chance of drawing a pair from a deck of multiple decks of cards?

If I had a shoe of multiple decks of cards, what's the probability of drawing 2 cards of the same value(pair)? E.g 2 of Spades and 2 of Clubs from a deck/shoe of 208 cards(4 decks) If I draw another ...
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Expected of number of discrete uniform variables whose sum is bigger than k (from characteristic function of discrete Irwin–Hall distribution?)

The problem Imagine we keep uniformly drawing $n$ integers $X_i$ from {0, 1, ..., 9} so that their sum is more than $10$. For instance, one draw would be {1, 0, 2, 5, 3}, hence $n=5$, and repeat this ...
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How to find the probability of winning this game?

Here's a very simple "game": Standard 52 card deck, shuffled. You deal cards face up, one at a time, while counting up towards 13, starting back at 1 once you reach 13. If the number you count out is ...
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2answers
634 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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What is the chance that someone is dealt a suit in the game of Bridge?

In a game of Bridge, what is the probability that some player has a complete suit? (There are four players in a Bridge card game. A deck of Bridge cards consists of 52 cards arranged in four suits ...
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1answer
74 views

Sampling inclusion probability for multiple items

Suppose we sample (uniformly, with replacement) $t$ times a set of $N$ items. What is the probability $x$ that the sample contains $y$ different items? This is representative of a real-life scenario ...
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49 views

What is the probability of 5 anomalies appearing in a batch of 100, from a data set of 1000 examples and 10 anomalies?

I am trying to work out the expected value of the following example: A data set contains 1000 examples in total. 10 examples can be considered anomalous. I am randomly drawing 100 examples to form a ...
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What is the probability of a achieving at random a log loss score within a certain interval

In a recent competition the participants where invited to predict the type of hemorrhage in 121,232 CTR images. More specifically, the participants were called to submit predicted probabilities for ...
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1answer
575 views

Computing number of people with the same name and date of birth based on sample data

A country has a population of 40 000 000 people. I have a sample with data for 1 500 000 people from this country. In this sample 1.9% of people have a "pair" (one or more) with the same first name, ...
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1answer
26 views

Number of all sample divisions in permutation test

I have a question with regard to the number of permutations in a permutation test. Assume I want to do a two group independent t-test. In the permutation variant of the t-test, I would repeatedly ...
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1answer
34 views

What is the intuition behind using binomial coefficient in the problem?

A box has 10 red coloured and 15 blue coloureds pens. If you were to pick out 7 pens (without replacement), what is the probability that you get exactly 2 red coloured pens? The answer is $$\frac{{10 \...
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Combinatorics and probability

I have 7 interviewees and 14 interviewers. Each interviewee is interviewed by 4 of those 14 interviewees. Is it possible to have every interviewee to be shared by at least 1 interviewer, as to allow ...

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