Questions tagged [combinatorics]

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

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Poker Calculator

Wrote a program to rate 2 hole cards (your hand) against a single random hand. Your two hole card are combined with a common 5 card board to create the best 5 card hand. You can play the board. ...
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Probability of observing all cards? [duplicate]

I have a stack of $n$ distinct cards. I randomly draw $N$ cards from the stack with replacement. Assume that $N \ge n$. What is the probability that I observe all cards at least once? I got this far:...
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Probability question about letters

Let's say we had the word STATISTICS, and it was hanging on a wall, then two letters in the word blew off, and somebody found them. What are the chances they put them in the correct order? here is ...
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1answer
42 views

Counting outcomes from arrangements of 4 bags

I'm having issues solving an introdutory exercise in a probability course. The following problem description is similar but not equal to the one I'm solving. I will first state the problem, then I ...
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95 views

Sampling pairs without replacement from two urns with overlapping populations

I have been trying to derive the coefficient of inbreeding lately, but I am at a loss. If I understand correctly it measures the probability of an allele "collision" i.e. getting the same allele from ...
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106 views

how do you calculate the number of combinations recursive partitioning uses in decision trees?

So I was wondering why random forests limit the levels of a variable to 32 levels, and I found an answer here : R's randomForest can not handle more than 32 levels. What is workaround? My ...
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122 views

Permutations in an Urn Problem, with Finite Number of Balls

In general I think I understand permutations fairly well. A simple example of permutations is drawing balls from an urn. I have 3 colors of balls, blue, green, and red, and I have 5 balls of each....
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1answer
75 views

Combination multiplication and joint probability of independent events probability

Is there direct correspondence between the rule of multiplication of combinations and probability multiplication in case of independent events? For combinations, whichever cup I choose, I can chose ...
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1answer
85 views

number of different events in a random sequence

I'm confronted with a probability problem that may have already been solved. I'm considering random sequences of m events. The number of possible events is n, and each type of event has the same ...
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2answers
42 views

Help with this combinatorial computation

I am going to run a test with N subjects. There are M=50 stimuli to be evaluated by the subjects but, for time and concentration reasons, a subject cannot evaluate all the stimuli. Therefore, I am ...
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1answer
606 views

Expected value of the largest number drawn in n drawings

An urn contains balls numbered 1 to N. Let X be the largest number drawn in n drawings when random sampling with replacement is used. (The event X k means that each of n numbers drawn is less than or ...
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probability of a digit repeating anywhere within a string [duplicate]

This is purely for interest. So I received from my bank a randomly generated (?) login code which is a string of 9 numerical digits (0-9). 5 of these 9 digits were the same number (but not in ...
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1answer
104 views

Probability of choosing all the numbers in a set

Suppose you have $m=5$ people and $n=4$ numbers. Each person chooses, independently and at random, one of these numbers. What's the probability that all the numbers will be chosen? I've been able to ...
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1answer
51 views

Looking for interdisciplinary research topic [closed]

TL;DR: Are there any areas that combine (1) statistics/ML, (2) computation theory, (3) set/type theory, and (4) real/functional analysis? I apologize if I did not accurately categorize the domains ...
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2answers
135 views

How to calculate the number of all combinations of all permutations?

I have a set n with m elements. I aim for the number of all combinations of all permutations of all elements of the set. An minimal example would be with a two element set [1,2]: All permutations of ...
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0answers
74 views

Fairness of a Coin. Number of heads vs probability of permutation

If I look at a sequence of 100 coin tosses and I get 90 heads the probability of the coin being fair vs not fair is a lot lower as answered in a lot of questions already. But when I go from looking at ...
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You randomly select two distinct integers between 1 and 100. What is the probability that the larger number is exactly twice the smaller number?

I recently took a HackerRank test for a Data Science position and got this question wrong. I came to 1/200. Here's how: There are 50 combinations that will make ...
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1answer
114 views

Calculating the expected time

Assume we are trying to crack a password. There are $2^{88}$ possibilities. I have a machine that can test $2^{40}$ passwords per second. What is the expected time to find the password if I use ...
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1answer
439 views

How to calculate the probability of an event occuring within n days, if we know the probability of it occuring per day

If an event can occur at most once per day, and the probability of it occurring is 10%. What is the upper-bound on the probability that the event occurs in an interval of L days of a month (assuming ...
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1answer
2k views

Dividing data into two groups as similar as possible

In this simplified example, I have five variables for each individual. The data is normalized, with mean 0 and sd 1 in each variable. I would like to divide this data into two groups with equal size ...
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1answer
39 views

confused with group testing

These are from notes,and I'm just very confused about what it tries to say Question 1: Does each test test all items? For example, if test 1 tested all items, test 2 tested all items etc, then why ...
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1answer
141 views

How to efficiently sample a large number of unique binary vectors

This is probably just a basic CS question, so I appreciate the help. I'm doing Monte Carlo permutation tests in matlab for One-Sample and Paired Two-Sample Tests (where there are x^n unique vectors ...
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1answer
65 views

Combination versus fraction

This happens to be about poker but I would like to ask it in a general way There are 47 cards remaining in the deck with 2 cards to come Of the 47 cards 9 make my hand - I only need 1 but 2 two ...
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1answer
1k views

Conditional probability - union of events

I'm going through Blitzstein's book Introduction to Probability and on p. 45 there is this exercise: Now, NOT using Bayes' Rule, $P(A|B) = \frac{P(A \cap B)}{P(B)}$. For the exercise, $P(A \cap B) = ...
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2answers
77 views

How to calculate the number of scatter plots?

if we have $p$ variables, the number of scatter plots we have is : $p(p-1)/2$. Why is this so? Is there any one who can explain this formula?
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1answer
29 views

Frequency of Item in Combination

Given a set of $m$ characters, if I pick $n$ of them, how many combinations contain a given character? A simple case: Picking two letters from ABCD has $\binom{4}{2} = 6$ possibilities, AB, AC, AD, ...
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752 views

How to efficiently choose $n$ subset out of a set of $m$ many numbers, in a randomized uniform manner?

Problems: It is fairly simple: we have a list of numbers $x_1, x_2, \ldots,x_n,\ldots, x_m$. Our goal is to randomly and uniformly choose a subset of $n$ many numbers out of these. This means that, ...
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917 views

Probability of five children in the same class having the same given name

On baby-naming forums, prospective parents repeat some version of their Fear of Jennifer all the time: "I don't want my child to be one of 5 in his class with his name." Thing is, no name comes even ...
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1answer
52 views

Proving that a dataset has all combinations of some features

What is the best way to prove / show that a dataset has all combinations of nominal values for a known group of features? For example, if attributes A and ...
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0answers
310 views

Entropy of independent draws of balls from a bin

Problem Setting Consider a bin containing $B$ balls, each ball is red or blue with equal probability ($0.5$). Let $R$ denote the number of red balls. It is easy to see that $R \sim Binomial(B,0.5)$ ...
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3answers
2k views

Number of combinations for a 4- digit code lock [closed]

Your suitcase has 4-digit code lock. You forgot the code, but you do remember that the sum of first two digits was equal to the sum of two second digits. How many combinations do you need to check in ...
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1answer
1k views

Sampling Distribution question - Probability finding

Suppose it is known that 8% of males are color blind. In a random sample of 500 males, what is the approximate probability that at least 10% of them are color blind? I am doing review for finals ...
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2answers
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3 vehicles out of 5 randomly showcased, what is the probability of each vehicle being showcased?

I have a basic probability doubt. If I have 5 different vehicles backstage, out of which 3 random can be showcased to the public. What is the probability of each vehicle to get to showcase? Once a ...
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70 views

Picking tuples from a list with low discrepancy

Given a list of m items, I am looking for a way to repeatedly pick a tuple of n distinct items from this list with low discrepancy. For example, suppose I have a list of 3d points, and I want to ...
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2answers
197 views

Permutations and probabilities for indistinguishable balls into indistinguishable fixed-sized buckets

I'm neither a statistician nor a mathematician so apologies if this question is asked and answered elsewhere or if I've misdescribed my problem. I am trying to figure out in a thread in stackoverflow ...
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1answer
93 views

Probability with Combinatorics

In a cellar there are 3 bottles of wine which are 18 years old, 4 bottles which are 20 years old and 3 more bottles which are 22 years old. Two bottles are being sampled by random. Let X be the ...
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0answers
2k views

How to find the maximum number of independent pairwise comparisons between groups

I have the following situation: Consider a dataset that is comprised of a number of factor levels x = gl(5,1) > [1] 1 2 3 4 5 > Levels: 1 2 3 4 5 I want to ...
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2answers
70 views

Pemutations and combinations problem

I have this one question in statistics: There are five women and six men in a group. From this group a committee of 4 is to be chosen. In how many ways can the committee be formed if the committee is ...
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Balls in bins with minimum spacing between filled bins

Say we have n identical balls and a row of m identical bins (with n<m). Say that the balls must be placed in the bins such that: No bin has more than one ball, and There are at least j empty bins ...
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1answer
232 views

How many times will a 12 letter word appear in a string of 1 quintillion random letters?

Let's say we think of a specific $12$ letter word. How many times on average will that particular word appear in a string of $1$ quintillion ($10^{18}$) completely random letters (i.e., uniform/equal ...
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1answer
54 views

Probability of getting two components in a graph

The following problem is an excerpt from the book Introduction to Probability Models, 10th edition, by Sheldon Ross. The origin of this problem is from $\S3.6.2$ A Random Graph, specifically after ...
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1answer
99 views

Alternating random number sequence combination [closed]

I have a problem that I have a hard time figuring out. We have a sequence of 10 numbers, 0 to 9. We want to find all the combinations where, when choosing from the sequence 18 times with replacement, ...
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2answers
77 views

Number of ways to put 9 different balls into 3 different urns to ensure nonempty urn?

All information needed is covered in the title. Suppose I have 9 different balls and 3 different urns. I want to put the 9 balls into 3 different urns. Calculated the number of ways to put the 9 ...
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27 views

Formal way to express the number of lists of k objects from n, having i unique elements

Say that I have a matrix of the $n^k$ ordered lists of $k$ objects from a supply of $n$, with replacement (which I am not sure how it's called). Note that $k$ may be greater, equal, or less than $n$. ...
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1answer
41 views

Counting the number of combinations from 5 equals sets

I have 5 sets with three elements each: A can be either 2,1 or 0 B can be either 2,1 or 0 . . E can be either 2,1 or 0 You can only pick one element from one of ...
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1answer
71 views

Need help calculating number of possible passwords with given criteria

I need help calculating the number of possible passwords with a given set of criteria. Here is the set of criteria: Passwords are case insensitive. Must be 6-14 characters. Must contain at least 1 ...
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4answers
185 views

Card deck shuffling, what is the probability that four lowest card of a suit are distributed to all four players?

In 4-player trick-taking game where taking a trick constitute a penalty (e.g. Hearts), I've often seen the card 5 (the fourth-lowest rank in a suit, when 2 is the lowest and Ace is the highest) takes ...
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1answer
48 views

I need to know how many combinations are possible from these, if I must select at least one (but can choose more than one) from each table

I have a very simple permutation question that I can't figure out on my own. So, I have {A, B, C} and {D, E, F} I need to ...
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2answers
73 views

Formulaic way to calculate the numerator of a probability mass function?

Problem description A package contains six batteries. Two of them are defective. Suppose that we randomly select three batteries and test them for defects. What is the probability that exactly one of ...
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1answer
54 views

About combinatorics

I have 5 Americans, 4 Mexicans, 3 Argentines and 2 Uruguayans. If I want to form a committee of 5 people, with the condition that each nationality is represented I know that I have to calculate the ...

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