Questions tagged [combinatorics]

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

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1answer
35 views

Randomly selecting integers with prescribed minimal distance and estimations

I am interested in sequences of $M$ distinct integers in $[[1,N]]$ (integers from $1$ to $N$) such that integers $I_m$ are separated by at least $\delta$ integers (taking into account the outer ...
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22 views

Drawing the same lottery numbers twince in a row - is it really different? [closed]

While for every single lottery draw the chance is always the same, I know that to get the very same result in the next draw, the probability is squared (1/probablity*probability). But I just cannot ...
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1answer
19 views

Optimizing production program

As part of the production of masscustomization products, I have to solve the following problem. A fixed set of sequences (changes from one product to another) is associated with a percentage value of ...
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1answer
27 views

Question in combinatorics

I would like to ask a simple question that happened to me on an interview. The question was very simple and it was as follows: We have 80 balls. We have 4 random groups (each group will have 20 balls)...
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19 views

Binomial coefficients - order matters in one subset of events

Question from 'Introduction to Probability' by Blitzstein-Hwang: Two chess players, A and B are to play 7 games. Scores are: a win is 1 points, a draw is 0.5 points and a loss is 0 points. a) How ...
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71 views

the probability of random subset

I'm trying to solve this puzzle but I get stuck. I thought about trying to use the law of total probability to solve intermediate problems with subset of size k but it didn't helped me that much. Is ...
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1answer
16 views

Why is the permutations of 6 objects where 2 are similar and the other 4 are similar is the same as 6c2?

I'm studying binomial expansion, and to get all the outcomes or arrangements of head appearing twice after throwing a coin 6 times without using a tree diagram, it's written that we can use 6c2 to get ...
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15 views

Expected Value of Total Number of Three Consecutive Colored Balls

Suppose we have 52 balls in random order where 26 are colored blue and 26 are colored red. The total count of a triplet of red balls in order or whenever a red ball directly follows two previous red ...
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Grouping classes of objects into more classes?

On page 15 of this Statistics textbook, "Mathematical Statistics and Data Analysis" by John A. Rice, the latter half of Example L has caught my attention: ...
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Which theory explains this estimation problem?

We have a bag with 100 balls, each labeled 1 to 100. Each ball has a color. 10 are green, 20 are red, 20 are yellow, and 50 are blue. Now someone continuously draws balls from the bag, telling you ...
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1answer
22 views

Permutations with exclusions - phone numbers problem

Problem from "Introduction to Probability" by Blitzstein-Hwang: a) How many 7-digit phone numbers are possible assuming phone numbers can't start with 0 or 1. b) Re-solve a) but now phone ...
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3answers
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How many ways are there to select exactly one heart in a hand of 5?

I was thinking about it in two ways: choose(13, 1) * choose (39, 4) = (13 * 39 * 38 * 37 * 36) / 4! Slot Method: 13 * 39 * 38 * 37 * 36 then divide out the repeats by dividing by 5! (number of ...
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1answer
32 views

Expected value of the largest item in a multinomial distribution

In this question, I use the notation on wikipedia. Suppose $X=(X_1,\ldots, X_k)$ follows a multinomial distribution with parameter $n,\mathbf p$, where $n$ is the sample size or the number of trials, ...
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1answer
20 views

Dependent Bead Draw

Copied below is a question that I recently tried to solve from a website: "There are 5 black beads and 5 white beads in your pocket. You pull the beads out of your pocket one at a time, at random ...
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Understanding combinations and down sampling strategies

Let’s say I have 27 definitions which can define a sample and I have 36 samples which must be defined. The order of these samples does not matter. For instance, one solution may be: 18 objects have ...
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Equation for no. of unique distributions of k objects, say balls, out of n total, comprised of multiple colors of varying amounts?

Say you have 4 orange balls, 3 red and 2 blue, and you take 6 at a time. I can manually that there are 9 possible distributions: 4o2r, 4o2b, 4o1r1b; 3o3r, 3o2r1b, 3o1r2b; 2o3r1b, 2o2r2b; 1o3r2b. But ...
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1answer
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Probability problem confusion

Hello i have a problem and I'm not sure why my solution is wrong. Problem There are 90 students. We're gonna split them in 3 groups, 30 students each. Find the probability that 2 given students A and ...
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1answer
28 views

What is the probability that you pick the winning numbers of lottery?

A state runs a lottery called “Pick 6.” A player buys a lottery ticket showing 6 of the numbers from 1 through 40. Then a set of 6 numbers is chosen as the winning numbers, and anyone whose ticket ...
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1answer
24 views

How to compute FN and TN explicitly?

Suppose that we have the following clusters ( output of some clustering algorithm ) based on 3 classes : Source : Source link The source explains how to calculate TP+FP and TP explicitly as following ...
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Lottery Machine Series Matching

Assume there is a lottery machine with 9 slots and 11 symbols (A,B,C,D,E,F,G,H,I,J,K) with decreasing probabilities. In this case, the probabilities are: [0.125,0.125,0.105,0.105,0.1,0.1,0.085,0.085,...
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1answer
28 views

sampling without replacement from a population

While counting number of ways to select a sample of size n, from a small(<=60 elements) population, we use combinations to count. For example, number of ways to select a sample of size 3 from a ...
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2answers
65 views

prove change in total probability of success in binomial distribution

A binomial distribution of $n$ samples and probability of success $p$ is defined as $ P(k) = \binom{n}{k} \cdot p^kq^{n-k} $. For a given value of $r$ where $r \in \mathbb{N} \quad \text{and } r \gt ...
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1answer
49 views

Hypergeometric distribution- problem with derivation

A random variable $K$ has hypergeometric distribution with parameters $N, m, n$, with probability mass function: $$ p_K(k)=\frac{\binom{m}{k}\binom{N-m}{n-k}}{\binom{N}{n}},\quad k\in\{\max(0,n+m-N),\...
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1answer
10 views

Combinations Question [duplicate]

There is a committee of 18 people, and a board of 5 people will be randomly chosen. What is the probability John, Jane, and Randy will all 3 be on the board of 5?
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1answer
14 views

Random shuffle of numbers

Suppose I have $N$ distinct numbers labeled $n_1, \cdots,n_N$. I want to obtain a random shuffle of the $N$ numbers. Are the following statements equivalent? $P$(card i at position j) = $1/N$ $P$(any ...
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23 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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14 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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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
33 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
15 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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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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2answers
47 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
105 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
27 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
26 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
42 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
50 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
51 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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40 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
49 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
49 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
74 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
45 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
46 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
18 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
17 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
59 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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7 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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20 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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