Questions tagged [combinatorics]
Counting or enumerating elements in a set or other finite discrete structure.
541
questions
2
votes
1
answer
31
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
...
- 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 ...
- 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 ...
- 71
7
votes
1
answer
203
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 ...
- 71
0
votes
0
answers
26
views
statistical interpretation to the infinite sum defining cosh(x) = sigma(x^2n/2n!)
the term $x^n$ / n! equals the probability of the sum of n uniform random variables between 0 and 1 to be smaller or equal to x. so $x^(2n)$ / 2n! equals the probability of an even number of random ...
- 31
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 ...
- 443
-1
votes
2
answers
171
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 ...
- 19
3
votes
2
answers
102
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 ...
- 31
3
votes
1
answer
87
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 ...
- 31
1
vote
1
answer
21
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 ...
- 11
1
vote
1
answer
49
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 ...
- 11
1
vote
1
answer
54
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:...
- 364
3
votes
1
answer
46
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 ...
- 225
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 ...
- 21
0
votes
0
answers
57
views
Lottery Numbers and runs test for randomness
Suppose we have a lottery with n winning numbers across m categories. If we take the results from N draws, and want to test the randomness using the runs test (testing on even/oddness of the winning ...
1
vote
0
answers
20
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 ...
- 321
7
votes
2
answers
241
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 ...
- 175
0
votes
0
answers
37
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 ...
- 111
0
votes
1
answer
38
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 ...
- 231
2
votes
1
answer
93
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:
...
- 23
3
votes
3
answers
119
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 &...
1
vote
0
answers
23
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 ...
- 11
2
votes
1
answer
63
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$
⚫⚫⚫ $...
- 260
0
votes
0
answers
28
views
Minimize within-group heterogeneity
Assume that I have a two-column table listing combinations of elements from two different vectors. A simple example using an R data frame x:
...
- 121
0
votes
1
answer
54
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 ...
- 3
0
votes
0
answers
42
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 ...
- 211
1
vote
0
answers
22
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 ...
- 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 ...
- 509
4
votes
1
answer
83
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{...
- 28.9k
0
votes
0
answers
16
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:
...
- 1
1
vote
3
answers
452
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)...
- 123
3
votes
2
answers
83
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 ...
- 68.3k
2
votes
2
answers
544
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 ...
- 485
1
vote
0
answers
47
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. ...
- 11
4
votes
1
answer
121
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 ...
- 91
1
vote
1
answer
59
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
$$\mathbb{P}[X=t]...
- 429
1
vote
1
answer
106
views
Maximize Mutual Information between multiple ("overlapping") RVs
I'd like to maximize the sum of Mutual Information between a RV $X$ and $K$ out of $N$ possible RVs $Z_i$.
$$ \max \sum_{i \in K} \text{MI}(X, Z_i) $$
However, when I unfold the sum I get
$$ \sum_{i \...
- 189
0
votes
0
answers
62
views
Counting outcomes in sampling with and without replacement from population
Suppose we have a population composed of 100 elements. Also, imagine we have a character with three categories A, B and C with proportions of 40, 40 and 20 in the population, respectively. We draw a ...
- 1
0
votes
0
answers
22
views
Binomial distribution and counting analogy
There is text completion question in an exam which has 3 blanks and for each blank there are 3 options , what is the probablity that we would choose the write combination of answer?
If I use Binomial ...
0
votes
0
answers
49
views
Question about how to think of the binomial distribution with a combination vs a permutation
I know there are many questions on why use a combination formula over a permutation formula for the binomial distribution ( for example here), but looking through multiple of these, I still don't ...
- 611
2
votes
0
answers
33
views
How much do we learn about a random subset? [closed]
Suppose we sample the following two random variables, for some large integer $n$:
Let random variable $X_1$ be a uniformly random subset of $m$ elements chosen from set $[n]:=\{1,\dots,n\}$, where $m ...
- 21
3
votes
2
answers
120
views
Book recommendations for Combinatorial Designs
I am well-acquainted to basic topics of experimental designs such as anova, ancova, crd, rbd, lsd, factorial design, robust design etc (as per the book of Dr. D. C. Montgomery). Now as per my recent ...
1
vote
1
answer
39
views
Probability of a person correctly guessing at least one number out of the two number another person chooses
Person A randomly chooses a number from 1 to 5 (inclusive) twice, so A ends up with 2 numbers chosen (can be the same number). Person B also makes a random choice from that list (only 1 number). What'...
- 91
2
votes
1
answer
43
views
Is there a formula for finding the number of ways 3 dice throws can add to 6?
Consider 3 dice throws: t1,t2,t3
Let E be the event that t1+t2+t3 add up to 6
What is the probability of event E?
I know the denominator is an example of ordered repetition. So with n=6 and k=3 there ...
- 641
1
vote
1
answer
58
views
How to use Stirling's formula of n! in this probability computations in random walk?
I want to compute $ \binom{2n}{n} p^n (1-p)^n = \frac{(2n)!}{n!n!}(p(1-p))^n, n=1,2,3...$
By using an approximation, due to Stirling, which asserts $ n! \sim n^{(n +\frac12)}e^{-n}\sqrt{2\pi}$
Where ...
- 1,204
1
vote
0
answers
88
views
Theory of 'Runs' and probability
I want to know how did author compute the probability of eleven runs if all arrangements are equally probable.
My attempt to answer my own question:
This example indicates wide applicability of the ...
- 1,204
3
votes
2
answers
102
views
What is the expected value of this process?
There are $n$ piles each containing $a_i$ stones. In a sequence of moves, Alex chooses two neighbouring piles randomly (containing, say, $A$ and $B$ stones) and combines them to create a single pile ...
- 31
0
votes
0
answers
27
views
All possible routes from 8 locations
I know that the combinatorics are simple, but I just can't remember and find a solution.
So I have 8 locations (A-H). It does not matter the direction. So Route A -> B is the same as B -> A. How ...
- 233
1
vote
0
answers
38
views
How many combinations of double sided cards can 4 judges select
The below scenario was taken from this video on the America's Got Talent show
4 judges, each judge picks two double sided cards. One card comes from a yellow pile and the other from a blue pile. Each ...
- 11
3
votes
1
answer
92
views
How to read the formal multiplication rule?
I have no problem understanding the question at Statistics Show How To
Question 3: A wedding caterer gives you three choices for the main
course, six starter choices and five options for dessert. How ...
- 641