Questions tagged [combinatorics]

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

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Calculate combinations when sample is equal but one can choose different samples

At a local lottery where I live you have to guess 5 out of 48 numbers and chances of hitting the jackpot (using nCr) are 1 in 1.712.304 (this is a small country), there is another type of play (way ...
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Determining if a die is fair or not by rolling six times and observing only the sum

Suppose you have two bags and each has 6 dice. In one bag all the die are fair. In the other each die has a bias - die “1” has a probability of returning 1 of $\frac{1}{6} +\epsilon$ and $\frac{1}{6} -...
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What is the probability that a random sample of 10 balls, taken from a collection of 50 blue balls + 50 red balls, has 5 blue balls and 5 red balls?

What is the probability that a random sample of 10 balls, taken from a collection of 50 blue balls + 50 red balls, has 5 blue balls and 5 red balls? When the balls are removed from the collection, ...
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Counting number of possible decision trees

I spent a lot of time on this question, and I search extensively on the internet, but I cannot find an answer (only an unanswered question here). Assume we have $p$ binary covariates, so that for each ...
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1 vote
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Forming triangles from 4n points on a square [closed]

Suppose that n points are given at any side of a square. None of these points coincides with a vertex of the square. How many triangles are determined by these 4n points? The answer given is like so: &...
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How do you make a generalized function for the probability of choosing a specific amount of special objects by swapping objects between groups?

I'll start by describing a very specific scenario. Suppose there are 15 distinct objects, 4 of which are special in some unknown way. I have to select 4 of the 15 objects simultaneously, not knowing ...
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2 votes
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Sum of values with different probabilities

Suppose I have the following linear expression: $S = x_1 + x_2+ \dots + x_n$, in which each $x_i$ can only assume the following values: -2, -1, 0, 1, 2 whose probabilities are 0.1, 0.2, 0.2, 0.25, 0....
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Is it possible to efficiently and uniformly sample from the partitions of a set that obey a given partial ordering?

I am having a problem similar to one that has been discussed before, how to uniformly sample from the set of all partitions of a set, but with a few differences. First, for my case I want to sample ...
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2 votes
1 answer
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How to determine the probability of a player getting 1st in a game, given win probabilities of head to head matchups of players?

I have predicted the probability that player A beats player B in a head to head matchup in a game, for all combinations of players. I would like to convert this into the probability of a player ...
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Probability that two samples from a multinomial distribution intersect in a specific way

The following combinatorially difficult problem has obvious applications in information retrieval and bioinformatics but I haven't seen its solution discussed anywhere. I am interested in computing ...
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Comparing the probability of co-occurrence between two pairs of vectors

Say I have two binary vectors $x,y$ of length $N$, with $n_x, n_y$ the number of $1$'s in those vectors and $n_{xy}$ the number of $1$'s that co-occur in the same spot. The probability that $n_{xy}$ ...
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Finding the probability of drawing all balls of the same color when there is a "wildcard" ball

I am working on a practice exam question for my Probability Theory class and am confused about a solution my professor gives us. The problem is as follows: There is a box with 16 balls in it, 4 red, 4 ...
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4 votes
1 answer
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How to find the probability of generating one specific number using a sum of five distinct numbers

I need help determining the probability/odds (decimal)/% chance of 'x' number appearing between 15-485, which comes from the sum of 5 randomly generated numbers (that range between 1-99). A specific ...
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Does adding Jokers into a normal 52 deck only add to the total # of possible hands, or does it also impact the existing hands' binomial coefficients?

The Jokers added to the deck in this case would be useless/junk cards, not wilds¹. My intuition is that there would be no impact to how we'd normally calculate the possible combinations/frequency for ...
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Probability of a similar sub-sequence of length X in two sequences of length Y and Z

Say we have two random sequences of length $y=3300$ and length $z=26$, where the sequences are made out of a combination of $k=4$ four letters ACGT. the letter in each position of the sequence is ...
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Stable sums from a set of data series

I have a set of daily data series spanning years of data representing different types of calls to a call centre. This data series are individually not constant, but I have been asked to find ...
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3 votes
1 answer
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How many sets of variables do I need to assess all pairwise correlations in a planned missing design

I have 54 variables that I need to correlate, but due to constraints I can only put up a maximum of 16 variables at a time for data collection. At the end of the day, I need estimates of the 1,431 ...
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3 votes
0 answers
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Expect Coin Value Problem ( a variant of coin change problem)

You have a wallet with four kinds of coins with the value of \$1, \$5, \$10, and \$25. Every kind of coin is unlimited. Every time you take money from the wallet, you have an equal chance of getting ...
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3 votes
2 answers
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Combinatorics - calculating probability of choosing n people from m groups of k people

I am trying to calculate and understand the following question: I have 10 groups of 10 people each and am choosing 10 people at random. What is the probability that all 10 people I have chosen are ...
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Why can't this (unordered with repetition) combinatoric problem be solved with this other approach?

I have the following problem: "There are 15 identical candies. How many ways are there to distribute them among 7 kids?" I know the conventional way to solve a similar problem (unordered ...
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Getting two probability distributions over graphs to agree on the probability of a given graph?

I have a simple fully connected graph with $N$ nodes and hence $\binom{N}{2}$ edges. I am asked to colour this fully connected graph, where I pick the colour for the nodes w.r.t a Multinomial ...
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Birthday Problem: Hypothetically Not Using the Complement

My Professor said that the probability that at least 2 of $k$ people have the same birthday if we didn’t use the complement is the sum of the probabilities that exactly 2 have the same birthday, plus ...
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Probability of 2 consecutive 6 in dice rolling experience using combinatorics

There's this question: You roll a dice three times, what's the probability of getting 2 "6" in a row? I have a solution for the problem that says: either getting 2 six in a row (first pair) ...
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Calculating chance of certain score in golf

Im trying to calculate the probability of getting a certain scores on a given golf course when i know the average score the player gets on each hole. I also know the distribution of eagle birdie par ...
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Formulas for Combinations of Colored Balls

Recently, I thought of the following question relating to "ordering colored balls according to some choice of constraints". Suppose there are 5 balls: Red Blue Green Yellow Orange Normally,...
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A more accurate way of inferring the pass/fail limits that were applied to a set of data?

This is best understood by considering measurements taken on each of a large number of integrated circuits on a semiconductor wafer at final test. Imagine that parameter 1 has a maximum and minimum ...
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1 vote
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Sampling identical sequences from two urns with unique colored balls

In very simple terms, my problem is as follows: given two urns, both with N balls where each ball has a unique color, I am interested in sampling sequences of N balls without replacement from each urn....
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combinatorics questions from coursera [closed]

Question 1 Imagine that now host have the following instructions. Put a prize behind a random door. Let the guest guess a door. If the guest chooses an incorrect door (with no prize), roll a dice (in ...
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Combinatorics correction: No. of ways to choose when unordered with replacement and its equivalent problem

Consider the problem: choosing 7 items out of 4 possible types A,B,C,D - with replacement and the order does not matter. There is a well-established formula for this and that is $n+k-1 \choose n-1$. I ...
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Calculating the amount of interactions with 3-way interactions

How do I calculate how many 3-way interactions could be tested when there are five predictors included in the model (multiple linear regression analysis)? And how many 5-way interactions could be ...
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Proving equivalence between two distributions?

I have a simple fully connected graph with $N$ nodes and hence $\binom{N}{2}$ edges. I am asked to colour this fully connected graph, where I pick the colour for the nodes w.r.t a Multinomial ...
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4 votes
1 answer
48 views

Probability of compatible love languages under uniform sampling

In Chapman's five love languages he claims that everyone has a primary and a secondary love language. Two people are said to be 'compatible' if their first love languages match, or the first love ...
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Why is chi squared for paired coin tosses so sensitive to slight deviations from 1:2:1?

I've been using googlesheets to calculate and understand how chi squares work in paired coin tosses as a model for pairing of alleles in Punnet squares. I used the randbetween(0,1) function to ...
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Constraint output for ML algorithm

When we have to select 1 output out of N outputs. We often have an ML architecture such as L1...
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8 votes
1 answer
126 views

Intuition about the coupon collector problem approaching a Gumbel distribution

The coupon collector's problem Let there be $n$ different types of coupons and we try to collect all of the types. We do this by independent random draws of coupons in which each type of coupon has an ...
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1 vote
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How can you estimate the actual number of options when there is some "fuzziness" involved? (Valve check/parking tickets)

I received a parking ticket the other day. By checking the valve positions the parking company claim that my car hadn't moved for a certain period. The protocol from the valve check shows that the ...
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1 answer
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How many ways to order such that one group comes first?

This question arose while playing a board game that uses cards. Imagine we have 11 cards total. 3 are red (R), 1 is blue (B), and 7 are yellow (Y). How many ways can we randomly order the 11 cards so ...
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1 vote
1 answer
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How do we know when order matters?

The following is from Ross "Introduction to probability and statistics for engineers and scientists" Example 3.5.4 We choose 6 pairs of room mates from a team of 12 basketball players. It ...
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1 answer
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A Pairwise Coupon Collector Problem

This is a modified version of the coupon collector problem, where we are interested in making comparisons between the "coupons". There are a number of constraints which have been placed in ...
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Calclulating the amount of duplication in a pool of random numbers [duplicate]

I want to figure out a way to determine how many unique numbers there would be in a given pool of random numbers. For example, lets say I generate 20 random numbers, each with a value between 1 and 20....
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4 votes
1 answer
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Probability of a draw between at least two options in an election with n voters and m options

I hope that this question has not been asked before, but i couldn't find an answer to it online or by myself. I came by that question during a project of mine: If i have an election with n voters and ...
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Schitt’s Creek David Ring Combination

So I’ve been trying to figure this one out. On Schitt’s Creek, David wears various combinations of rings. He has four rings and he can arrange them on his ten fingers in various ways - each ring is ...
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Chances of four people being drawn together

There are eight people in a golf group. There is going to be a draw to split them into two groups of four. There are four people within the group who would like to be drawn together. What are the odds ...
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1 answer
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Probability of not picking a row in a random draw where the number of rows are N

There are $N$ rows :$R_1, R_2,R_3,..., R_N$. What is the probability of not picking a row in a random draw? My try and understanding : Let $X$ be a random variable which is defined as follows: $$X = \...
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1 vote
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Inferring Sequence of Distinct Objects From Historical Data

I have K distinct objects that are stocked at a shop every day. I'd like to use data from prior days to predict the order in which items will be sold today. Is there a straightforward way to predict ...
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0 answers
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Statistical approach for football competition

There are 10 teams in a club. Each week, there are 3 games where each team plays another team from the same club. All games are played in the same park. The park has 5 pitches. There are 5 teams that ...
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Estimating true and false matches from record linkage between two datasets

Suppose I have two datasets I want to join. Dataset A keeps track of users of a local health care service, identifying them by an unique key. Dataset B records all hospitalization events from citizens ...
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2 votes
1 answer
171 views

Runs test based on the median intuition

The following document http://www.ifp.illinois.edu/~ywang11/paper/ECE461Proj.pdf contains a very nice summary of some nonparametric tests for randomness. My question concerns the popular Runs test ...
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Expected Value of a Fundraiser Raffle Ticket with Three Prizes

The raffle ticket above represents a local fundraiser. The first prize is a smart TV valued at 1200 dollars, the second prize is a cooler valued at 200 dollars, and the third prize is a 100 dollar ...
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-1 votes
1 answer
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Match a Sequence to a bigger Sequence

I am trying to solve the following problem: Say I have a small sequence $\sigma = (σ_{1}, σ_{2}, ..., σ_{i}, ..., σ_{N})$ and a larger $\hat{\sigma} = (\hat{σ}_{1}, \hat{σ}_{2}, ..., \hat{σ}_{j}, ..., ...
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