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Questions tagged [combinatorics]

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

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4 votes
2 answers
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Making 3 comparisons with a ternary variable

I have a study in which I want to study the effect of a ternary variable $x$ in interaction with a binary variable $y$. The ternary variable is categorical, and for some reasons, I want to do the $3$ ...
Alexandra Steinhilber's user avatar
1 vote
1 answer
40 views

10 identical socks, 7 drawers: What is the probability that at least one drawer contains 3 or more socks?

My nephew asked me the question thinking I would answer him very quickly, but I got stuck until today..... There are 7 drawers and 10 socks, all the same. The socks are randomly distributed in the ...
Xavier Labouze's user avatar
0 votes
0 answers
25 views

Combinatorics question

Let me preface this by saying that this might be a trivial question for some of you. I want to find a formula that will help me automatically calculate the number of occurrences of certain kind of ...
Doktor's user avatar
  • 1
0 votes
1 answer
65 views

How to solve this "easy" probability problem

I'm struggling on how can I solve the following problem. I tried with a lot of calculus without success, but i think that the way to solve is: $$P(\text{at least one failure})=1−P(\text{no failures in ...
Ga13's user avatar
  • 280
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0 answers
12 views

What is the Pseudo dimension for Lipschitz function class?

I am trying to get a bound on covering numbers with $L_2(P_n)$ norm using pseudo-dimension or fat-shattering numbers for Lipschitz classes.
fun yaaar's user avatar
1 vote
1 answer
44 views

Supposed to be a simple question about entropy

Let say there is an urn which contains balls of different color. It is a well known formula to calculate entropy of balls in the urn: $H = - \sum P_i\cdot\log(P_i)$ where $P_i = \frac{M_i}{N}$, where $...
irriss's user avatar
  • 121
5 votes
1 answer
152 views

Statistical tests for data on occurrence of reptiles found under different types of artifical cover objects

I am studying small-bodied ground snakes and skinks (lizards). I use artifical cover objects to determine their presence. Artificial cover objects (hereafter CO) consisting of 7 different types of ...
BAE's user avatar
  • 53
4 votes
1 answer
84 views

Conditional expectation of Poisson, conditional on Poisson sums

Consider independent Poisson random variables $X_1\sim \text{Poisson}(\alpha_1)$, $X_2\sim \text{Poisson}(\alpha_2)$, $Y\sim \text{Poisson}(\lambda)$, and suppose $Z_1=X_1+Y$ and $Z_2=X_2+Y$. I want ...
Car Loz's user avatar
  • 850
1 vote
0 answers
35 views

Enumerating feasible solutions to the subset sum problem using Gibbs sampling

Given a set of $m$ strictly positive real numbers $W = \{ w_{1}, \dots, w_{m} \}$, I want to find subsets of $W$ whose sum is less than or equal to a maximum value $N$ using Gibbs sampling. To do this,...
scj's user avatar
  • 101
2 votes
1 answer
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Combinatorics: create a 3-digit number out of 2 digits?

We have a set of two numbers {1,2} an we wish to create a 3-digit number out of the two numbers without order mattering. Here is my approach since there are 2 values for each spot then it's 2^3 since ...
manylya's user avatar
  • 23
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0 answers
63 views

Probability of 5 card texas holdem poker royal flush

How would I calculate the probability of getting a royal flush with 2 player Texas holdem poker. A single randomly shuffled 52 card deck, 2 players (p1 & p2) are sequentially dealt 4 cards i.e. p1 ...
Jimmy3421's user avatar
1 vote
0 answers
102 views

Probability problem - Jackpots

The rules of the final round of King's Lottery in Byteland are simple: The player starts to draw balls from an opaque bag containing $a$ red balls, $b$ green balls and $c$ blue balls initially. After ...
Amir Alakbarli's user avatar
3 votes
2 answers
83 views

What is the probability of being dealt two pairs when all $\binom{52}{5}$ poker hands are equally likely?

Question from Sheldon Ross's First Course in Probability, Chapter 2: If it is assumed that all $\binom{52}{5}$ poker hands are equally likely, what is the probability of being dealt two pairs? (This ...
Roy's user avatar
  • 31
1 vote
0 answers
58 views

Generate unbalanced sample [closed]

Suppose we have $N$ individuals consisting of two different groups, A and B. Each group contains $N/2$ people. The label of each individual $Z$ follows a binary distribution with probability $P(Z=A)=P(...
Fangzhi Luo's user avatar
0 votes
0 answers
14 views

Conditionally Quasi-random Choices and Qualitative Dependent Variable Models

I have a scenario about limited dependent variable models and some of their implications. Suppose that I estimate some limited dependent model, it does not really matter which one, that captures the ...
DarkenExcalibur's user avatar
2 votes
1 answer
36 views

How would I formalize this combinatorics question?

We have workouts, durations, and difficulties and we want to know how many unique combinations of these items we can form. Workout Duration Difficulty Strength 10mins Beginner Yoga 15mins ...
deblina's user avatar
  • 21
2 votes
0 answers
31 views

Probability of rolling any number 10 times at any given point in the sequence [duplicate]

Imagine a race, where each runner has a number, the race track is 100 meters, and each time you roll a fair 6-sided die, the corresponding runner moves 10 meters. I would like to calculate the ...
Bat_Dad's user avatar
  • 21
0 votes
0 answers
6 views

Finding combination of bacteria that represents maximal functions encoded by them

I have the following data where rows represent functions and columns represent bacteria. I have 18 bacteria with a total of 160 functions (all funcitons are present in at least one bacteria). I want ...
Arijit's user avatar
  • 71
7 votes
1 answer
216 views

Amount of total valid strings with multiple constraints

There is a string that has five potential letters (A,B,C,D,E). However, we have some constraints: Each unique ordering of the string is a different passcode. Letters can be used more than once or not ...
Lauren Xu's user avatar
0 votes
0 answers
12 views

Number of permutations when there's a pair of identical items [duplicate]

Suppose there's a bag with four balls; color 0 appears once, color 1 appears once, color 2 appears twice. How many ways are there to draw these balls? Having done this by hand, I believe there's 12 ...
Arturo Sbr's user avatar
-1 votes
2 answers
172 views

How many distinct possibilities for the first $k$ elements of a permutation of $n$ items, when some of the items are identical?

Problem If I have to rearrange $n$ items (which some of them are identical), taking the order into account and only selecting the first $k$ items of every permutation, how many permutations do I have ...
georgiem's user avatar
6 votes
2 answers
137 views

Combinatorial/probabilistic meaning/analogy for $x^n / n!$

The expression $x^n / n!$ appears in the infinite sum defining $e^x$ and similar terms in the sums defining $\cos(x)$, $\sin(x)$, etc. I would like to know if there is some combinatorial/probabilistic ...
Gilad's user avatar
  • 61
3 votes
1 answer
105 views

Number of configurations in which you can roll 2 6's in a row

A colleague at work gave me a problem to think about. If you throw $n$ dice, how many configurations are there where you get at least 2 6's in a row? I worked a lot with Pascal's triangle and in the ...
Francesco's user avatar
1 vote
1 answer
45 views

How do I create random teams of people, each person has multiple parameters, and the parameters are equally distributed among teams

As the title suggests, I am trying to divide a group of people randomly into different teams of equal size. However, each person has parameters that go with them (such as age). If there is only one ...
Stephen's user avatar
  • 11
1 vote
1 answer
52 views

Question on permutation and combination

I am having a trouble understanding how to answer the following question and what method to use. The final answer is $415800$ Suppose there are 4 committees A, B, C, and D. 11 candidates are randomly ...
Rudinberry's user avatar
1 vote
1 answer
89 views

Run lengths in a binary string

Among the binary strings of length $n$, what is the distribution of the lengths of the homogeneous runs ? E.g., for $n=4$ the possible strings and run lengths are $$0000: 4;0001: 1,3;0010: 1^2,2;0011:...
Yves Daoust's user avatar
3 votes
1 answer
47 views

Optimizing Treatment Group Assignments

When setting out experiments, I want to make sure treatment groups are as balanced as possible. Instead of using randomization, I've started to use the following process. I first collect some ...
David Moore's user avatar
0 votes
0 answers
75 views

Stochasticity challenge

If a string of characters (S) that is 325384 long, contains 458 As and 22 Bs. What is the probability that if the 458 As and 22 Bs were randomly positioned along the 325384 string of characters that ...
Duranimo's user avatar
1 vote
0 answers
27 views

Probability of seeing an ace before seeing a two in a 52 card deck?

We have a shuffled, standard 52 card deck with 4 Aces and 4 twos. I am confused about the probability of seeing an ace before seeing a two. By symmetry, it seems obvious that the probability should be ...
wwyws's user avatar
  • 321
7 votes
2 answers
335 views

Flipping an unfair coin until there are more heads

I came across this question on Quora. You have an unfair coin for which heads turns up with probability $p=\frac 35$. You flip the coin repeatedly until there have been more heads than tails. How ...
Nothing special's user avatar
0 votes
0 answers
38 views

Probability of not detecting fire if two of the sensors don't work well

I am trying to solve the following problem of probability: "In a certain factory, there is an alarm system composed of 5 sensors that work independently to detect fire. The probability that a ...
CharlesJA's user avatar
  • 111
0 votes
1 answer
43 views

Number of ways you sit next to your friend with 4 other people at a round table [closed]

This is a question I have after seeing Q1 of this question. There are a total of 6 people including you and your friend, and you guys are about to sit at a round table. The method of counting the ...
user21's user avatar
  • 231
2 votes
1 answer
127 views

$n$ people each randomly assigned a number from $1$ to $m$ with replacement. Probability that exactly one number is assigned to more than one person?

Suppose that $n$ people are each randomly assigned a number from $1$ to $m$ with replacement. What is the probability that exactly one number is assigned to more than one person? What I have tried: ...
Brandon's user avatar
  • 23
3 votes
3 answers
141 views

Deterministic formula for average number of unique items picked

I am curious how to formulate a deterministic answer to a problem I have in mind. I have computed it stochastically, but am unsure of how to frame and compute the problem deterministically. My prob &...
Jason Morrison's user avatar
1 vote
0 answers
28 views

Given a set of random variables, how can I find a linear combination of these variables satisfying a constraint on the sum of their permuations?

Say I have n random variables, {X0...Xn}, n>9. I also have another set of random variables constructed from the first set, where each of these are the sum of 9 ...
Eli Katz's user avatar
2 votes
1 answer
66 views

What is the relation between permutation and combination in ${}_nC_k$?

Suppose we flip 3 coins. The possible outcomes can be pictured, with heads in black and the number of heads denoted $k$, as: ⚪⚪⚪ $k=0$ ⚪⚪⚫ $k=1$ ⚪⚫⚪ $k=1$ ⚫⚪⚪ $k=1$ ⚪⚫⚫ $k=2$ ⚫⚪⚫ $k=2$ ⚫⚫⚪ $k=2$ ⚫⚫⚫ $...
RodParedes's user avatar
0 votes
1 answer
78 views

Non-transitive dice probabilities

I got the following question on non-transitive dice, as shown in the following picture. Suppose you roll two white dice against two red dice. What is the probability that the sum of the white dice is ...
ezrix's user avatar
  • 3
0 votes
0 answers
43 views

Counting Problem, Correct Solution?

A committee of 5 people is to be selected from a group of ten. How many are there if among the group is a married couple which can only be selected together. So there are ${10 \choose 5}$ committees ...
user2550228's user avatar
1 vote
0 answers
29 views

Probability of net gain being greater than 0 lottery question

I'm working on some practice exams and I came across a probability question that stumped me. "In a lottery game, for any lottery ticket bought at random, the chance of winning each prize is ...
carter's user avatar
  • 11
1 vote
1 answer
84 views

Combine probabilistic forecasts with weighting

Suppose I want to compute the probability that Argentinia wins the worldcup semifinal Argentinia - Croatia. I have two independent sources of information about this probability, whereby source A says ...
monade's user avatar
  • 519
5 votes
1 answer
100 views

How to calculate the total number of orderings of two samples while allowing for ties?

The total number of orderings of a sample (from lowest to highest) of two independently distributed continuous variables $x$ and $y$ is calculated thus: $$\text{No. of orderings } = {n_x +n_y \choose{...
Alexis's user avatar
  • 30.3k
0 votes
0 answers
17 views

What statistical methods can I use to determine the popularity of item combinations taken from two groups?

Let's say I have two tables, listing projects with the programming languages and frameworks they use: ...
Anais's user avatar
  • 1
4 votes
3 answers
462 views

Why is the number of subsets in a sample space $2^n$?

I am working my way through the statistical inference textbook and am stuck on this question: Suppose that a sample space $S$ has $n$ elements. Prove that the number of subsets that can be formed ...
Solal's user avatar
  • 71
1 vote
3 answers
1k views

What is $P(A|B,C)$ when $B$ and $C$ are independent?

Let's say that I know the following: $P(A|B)$ is the probability that a storm is coming given it's cloudy. $P(A|C)$ is the probability that a storm is coming given that the dogs bark. $P(B)$ and $P(C)...
William Grand's user avatar
3 votes
2 answers
98 views

How many rounds to visit every place on a Monopoly style game board

I see some kids playing on a circular gameboard trying to visit every field and I wonder how long it will take before they finish. I can simulate it like below, but is there possibly an elegant ...
Sextus Empiricus's user avatar
3 votes
2 answers
2k views

Parking lot problem

I have found similar questions in these links (1,2,3), but none of them is from CrossValidated (and I still don't grasp the solution, so here I am). The problem is this: Tom and Jerry park their cars ...
Chris's user avatar
  • 525
1 vote
0 answers
65 views

Determining Combinations of Variables Values that lead to a certain outcome

Say I have a dataset with the following variables: age, gender and Impact. Impact is the dependent variable and it's binary, age and gender are independent variables numerical and binary respectively. ...
Euler's user avatar
  • 11
5 votes
1 answer
144 views

Probability of winning the series given not playing the first game

Suppose three pool players A, B, C are equally good, meaning that they each have a 0.5 probability of winning against each other. They are playing a game as follows: the winner of a game plays whoever ...
Oscar Wan's user avatar
  • 101
2 votes
1 answer
76 views

Weird definition of negative binomial distribution

In a paper I am reading, they define the negative binomial as the following: random variable $X$ has a negative binomial distribution with parameters $p \in (0,1),k \in \mathbb{N}$ if $$\mathbb{P}[X=t]...
jacob's user avatar
  • 459
2 votes
1 answer
163 views

Maximize Mutual Information between multiple ("overlapping") RVs

I'd like to maximize the sum of Mutual Information between a RV $X$ and $K$ out of $N$ possible RVs $Z_i$. $$ \max \sum_{i \in K} \text{MI}(X, Z_i) $$ However, when I unfold the sum I get $$ \sum_{i \...
N8_Coder's user avatar
  • 219

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