Questions tagged [combinatorics]
Counting or enumerating elements in a set or other finite discrete structure.
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0answers
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 ...
0
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0answers
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 ...
0
votes
1answer
16 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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0answers
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. ...
0
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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 ...
1
vote
2answers
75 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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0answers
18 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.)...
...
1
vote
3answers
70 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 ...
0
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2answers
47 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 ...
0
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0answers
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, ...
2
votes
1answer
31 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
11 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!...
0
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1answer
235 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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0answers
29 views
Probability of having a match in an existing database
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 ...
3
votes
0answers
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 ...
0
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1answer
26 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 ...
0
votes
1answer
53 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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0answers
21 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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0answers
21 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 ...
0
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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 ...
2
votes
1answer
35 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 ...
0
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0answers
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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0answers
28 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 (...
4
votes
1answer
103 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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0answers
17 views
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 ...
2
votes
1answer
89 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
...
0
votes
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 ...
0
votes
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 ...
4
votes
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 ...
2
votes
2answers
129 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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0answers
143 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 ...
1
vote
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 $...
0
votes
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. ...
0
votes
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....
2
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1answer
39 views
Bridge Hand Probabilities
Each of the 4 players in the game of bridge get dealt 13 cards. One player and his partner can see they hold 8 of the heart cards so they know that the 2 remaining hands they can't see hold the ...
2
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0answers
32 views
What is the name of this test for the probability of differences between lists and is it valid?
And are there any better methods?
In a 1969 book about investing, the author Burton P. Fabricand describes a method for assessing the effectiveness of a stock picker. Ask the picker to provide a list ...
3
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1answer
94 views
Different ways of counting the same combination
What’s wrong with this reasoning:
How many different ways to pick a team of 3 from 4 people? ${4 \choose 3}$
Alternative way of counting is to choose a team of 2 from the 4 first (${4 \choose 2}$ ...
0
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0answers
45 views
Realisation of at least one event among N events
Two dice are thrown r times. I want to find the probability $p_r$ that each of the six combinations (1,1),...,(6,6)appears at least once.
My Answer:-$p_r=6*(1-(\frac{35}{36})^r)-15*(1-(\frac{34}{36})^...
2
votes
3answers
569 views
Coin Tossings probability
I want to find the probability that in ten tossings a coin falls heads at least five times in succession. Is there any formula to compute this probability?
Answer provided is $\frac{7}{2^6}$
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0answers
28 views
Predicting a combination
Question
Suppose we have a training set of families. Where each family is defined as such…
Family: A list of integers. Each integer is the age of one of the family members.
(e.g. with a 45 year ...
0
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0answers
23 views
Optimizing a sequence of events on a variable conditional on the event
I'm trying to understand how I would approach a problem aiming to optimize a sequence of events to maximize another variable. For example, say I start selling a t-shirt at multiple stores, where each ...
2
votes
1answer
122 views
Feller shoe-matching problem
This problem have been taken from the book' An Introduction to Probability Theory and Its Applications' by Williams Feller(1906-1970)
Note:- Assume in each case that all possible arrangements have ...
0
votes
1answer
29 views
Randomly choose 50% of n things, but only 1% of specific k things implies what?
Summary: If you randomly choose 50% of n entities total, but it turns out you only chose 1% of k specific pre-chosen entities, does that imply n is likely to be much larger than k?
I'm trying to use ...
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1answer
44 views
Probability of single entry originating from same individual
If I have a bag with n balls of k colors (equal number of each color), and I draw x balls (with replacement), what is the chance of getting two of the same color?
I tried to simplify but not sure if ...
2
votes
1answer
38 views
Difference between permutations and combinations
Ok, I know this question has been asked a thousand times. I do understand that permutations mean that order matters where combinations not. I know there's more to it, like what we are counting and the ...
1
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0answers
47 views
Probability that 3 out of 4 sets have non-empty intersection
I have $4$ randomly sampled subsets from population set $S = \{1,2,...,100\}$. Each subset has size $24$.
What is the probability that at least one element is common among $3$ out of $4$ subsets?
5
votes
2answers
56 views
Combinations or Counting general formula
Assume you have 5 numbered bins and you have 2 red balls, 2 blue balls and 2 green balls (You can assume that there are equal number of red, blue and green balls). I am trying to derive a general ...
0
votes
1answer
118 views
Does an approximation exist between the Cumulative Binomial Distribution and the probability of combinations?
Suppose that we have two disjoint subsets $A$ and $B$ of $Z$. Both $A$ and $B$ have a large number of elements each.
We want to compute the probability of picking $k$ elements from $A$ e $B$, with ...
1
vote
1answer
109 views
Why do we use ${n\choose k}$ for a binomial distribution instead of ${n+k-1\choose k}$?
I am trying to get my head around this. In my understanding a binomial distribution uses replacement and ${n\choose k}$ precisely states that there's no repetition and that's not the case with a coin ...
