Questions tagged [combinatorics]

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

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1answer
30 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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0answers
13 views

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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22 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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2answers
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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1answer
96 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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1answer
22 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 a time ...
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1answer
19 views

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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1answer
39 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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0answers
49 views

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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2answers
41 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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17 views

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
28 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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1answer
41 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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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
40 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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0answers
14 views

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
49 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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5 views

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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0answers
19 views

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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0answers
13 views

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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1answer
14 views

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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0answers
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
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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0answers
33 views

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
209 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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0answers
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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0answers
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
45 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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2answers
61 views

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
80 views

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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0answers
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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0answers
20 views

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
25 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
33 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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0answers
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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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0answers
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Passive Combination of two Classifiers at the level of Class Labels

I have three classifiers - 1. Vision based classifier trained to detect class labels such as pedestrian and cars 2. Radar based classifier to detect same class labels 3. Lidar based classifiers to ...
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1answer
38 views

Calculating the possible number of configurations

I am wondering how did they get the $19200$ possible configurations? Like, $5^6 = 15625$, where $6$ is the number of hyper-parameters: ps: just to check I'm doing the righ thing. Is this okay? The ...
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1answer
78 views

Acronyms duplicates (a generalisation of the birthday problem)

I brought more precision into the way I ask this question to make it investigatable: Assume that everyone has a First name, a Family name, a Street name and a Suburb name. If you make a four-letter ...
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1answer
44 views

How to generate a conditional subset?

I want to develop an algorithm to generate a random subset (size $k$) from $\{1, . . . , n\} $ given that it contains at least one of the elements in $\{1, . . . , s\}$ ($s,k\ll n$). This is what I ...
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1answer
100 views

What is the probability of getting three pairs that makes up seven when six dice are rolled?

Stumbled upon this question in a game of dice where a point is dealt for each pair that makes seven. In the game there were a total of six dice that were to be thrown up to three times. Each time a ...
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0answers
7 views

Expected number of random permutations to see every permutation

Suppose I have n objects and I randomly permute them until I've seen every possible sequence of those objects. What is the expected value of the number of trials it would take me to get to that point?
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1answer
129 views

Basic question about combinatorics and probability

For the following question: Six race cars, each a different color, race each other around a track. If the order in which they finish is random, what is the probability that the turquoise car ...
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3answers
140 views

Is there a formula to calculate all possible unique permutations of n elements over p positions?

Is there a formula to calculate all possible unique permutations of n elements over p positions? Please imagine the following ...
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1answer
84 views

Rental truck probability confusion

would appreciate if anybody could help me with this probability problem A rental truck agency services its vehicles on a regular basis, routinely checking for mechanical problems. Suppose that the ...
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1answer
32 views

How can I generate a completely balanced combination

Attaching a code that generates unique combinations of 8 letters but, it still lacks a condition wherein all letters should have equal counts per column. For my code, basically, letter a to h should ...

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