Questions tagged [combinatorics]
Counting or enumerating elements in a set or other finite discrete structure.
0
votes
1answer
32 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 ...
1
vote
0answers
26 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
57 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
votes
1answer
25 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
50 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 ...
1
vote
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
18 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
votes
2answers
30 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
29 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 ...
0
votes
0answers
27 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
100 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
16 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
76 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
73 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
127 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^...
1
vote
0answers
142 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
19 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
22 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
votes
1answer
35 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
votes
0answers
31 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
votes
1answer
93 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
votes
0answers
44 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
567 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}$
0
votes
0answers
27 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
votes
0answers
18 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
117 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 ...
1
vote
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
36 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
vote
0answers
46 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
55 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
106 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
108 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 ...
1
vote
2answers
107 views
Probability of substring inside string of coin-tosses
Let's say we flip a coin $N=100$ times and get "HTHTHTHHHTHHHHTTHTT...HHT". What is the probability that se letter combination "HHHHH" occurs at least once in this string of $100$ letters?
If not a ...
1
vote
1answer
30 views
finding expected value when a set is given and a subset of size n is chosen
I found an interesting coding challenge on pramp by a friend but I couldn't do it in time. Anyhow, it says given a set { 3,14,7,22,29,33} and random 3 element subset is generated each time and its ...
2
votes
0answers
68 views
Probability of a contingency table with given marginals
In Fisher's exact test we consider all contingency tables $(n_{ij})_{i,j=0,1}$ consistent with given marginals $(n_{0.},n_{1.},n_{.0},n_{.1})$, where $\sum_i n_{i.}=\sum_j n_{.j}=N$.
Each table has ...
0
votes
0answers
10 views
Number of ways to distribute n not necessarily distinct objects into r identical boxes?
I'm wondering about the number of ways of distributing n (not necessarily distinct objects) into r identical boxes.
I tried using Stirling and bell numbers to calculate the result which works ...
2
votes
1answer
49 views
Number of combinations when swapping two sets with some elements in both
This question is closely related to this other one: I have two sets and I want to know the number of possibilities I can do with the elements of these two sets.
As a possibility I mean changing ...
1
vote
1answer
36 views
Permutation and counting techinques
A bookshelf has 15 books. in how many ways can 4 books be removed such that no two adjacent books are chosen?
I started to solve the question by saying that the first book can be slected in 15 ways, ...
0
votes
1answer
26 views
Negative Binomial Coefficients -nCk = (n+k-1)Ck
I am unable to understand that why $\binom{-n}{k} = \binom{n+k-1}{k}.$ Please help me in understanding this.
0
votes
0answers
21 views
Distributing elements over multiple buckets of a varying size
If one has b buckets that each have a size of b_i and one has to place n elements in the buckets. How can I mathematically described the amount of different permutations the elements can be placed in ...
3
votes
1answer
72 views
Coupon Collection Random Sum [closed]
I have a problem which involves the standard coupon collector's problem to find a probability density from the generating convolution. I start by defining the problem and a few basic statistics. Let ...
1
vote
1answer
41 views
Problems involving Conditional Probabilities and/or Combination
In a poker hand consisting of 5 cards, find the probability of holding 3 aces.
Solution A: $$\frac{^{4}C_3 \times\ ^{48}C_2}{^{52}C_5}$$
Solution B (Does not work):
$$\frac{4}{52}\times\frac{3}{51}...
-1
votes
1answer
23 views
Variance from sampling from a collection of marbles
Suppose I have $N$ marbles, $k$ of which are black. Let $X$ be the number of black marbles obtained from randomly choosing $M$ ($\leq N$) marbles. What is the variance of $X$?
Obviously if $M=N$ then ...
1
vote
0answers
33 views
Arrange features in a table that minimizes covariance
I have a dataset of vectors represented by a $n_{rows} \times n_{cols}$ matrix $M$. Each row is a vector and each column is a feature. There are $n_{cols} = n \times n$ features.
We use the sample ...
1
vote
1answer
721 views
Python library for combinatorial optimization
I've been recently working with a combinatorial optimization problem defined as follows. Given two sets of items, A and B, select the best combination of these items given a scoring function $f(A,B) \...
