# Tagged Questions

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### How to convert a problem to a stars and bars problem?

Continued question from here. How can I convert a problem like: $$\def\x{x_}\x1+\x2+\x3+\dots =15$$ with $\x1\leq3$ Back to a simple stars and bars problem such as $$y_1+\x2+\x3+\dots=33$$ How ...
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### probability rolling a dice 5 times

I can't solve this problem: What is the probability that, when rolling a dice 5 times, the number of times when you get a 1 or 2 is greater than the number of times when you get a 6. any help?
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### Expected frequency of most frequent die roll

Suppose we have an fair $m$-sided die, and we roll it $n$ times. What is the expected frequency $E(n, m)$ of the most frequently rolled face? If we fix $n$ we can calculate $E(n,m)$ like so. Let ...
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### Collision of 8 Digit, Base-36 Numbers

I have an algorithm that generates a random 8 digit, base 36 number with uniform distribution. Thus, this algorithm can generate $36^8$ unique numbers. I run my algorithm 10,000 times, and write ...
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### Probability: Linear Seating Arrangement

Okay, I'm new at probability and statistics, so please try to answer this as thoroughly as possible and explain why you did everything, from using a specific number to why using factorials and ...
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### How to use stars and bars(combinatorics)

How to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$ Where $x_i\in\mathbb{N}$ Is this the correct time to apply the method?
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### Probability of dying from smallpox?

A family of four is infected with Variola major. There is a fatality rate of 30%. Calculate the probability that... Here are my attempts, The probability that nobody dies, ...
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### Boxes and colored balls with replacement

Suppose there are $n+1$ boxes numbered from $0$ to $n$. The $i$-th box contains $i$ white balls and $n-i$ black balls. A box is chosen randomly and a ball is selected from the box, after that the ...
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### Pólya's urn scheme, proof using conditional probability and induction

Problem An urn contains $B$ blue balls and $R$ red balls. Suppose that one extracts successively $n$ balls at random such that when a ball is chosen, it is returned to the urn again along with $c$ ...
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### Dice, balls and boxes probability problem (conditional probability)

Problem Suppose there are two boxes $A$ and $B$ such that $A=\{\text{5 red balls and 3 white balls}\}$, $B=\{\text{1 red ball and 2 white balls}\}.$ A dice is thrown, if the result is $3$ or $6$, a ...
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### Colored balls in three boxes (conditional probability problem)

Problem Suppose there are three boxes numbered with twenty balls in each of them. The first box contains twenty white balls; the second, fifteen, and the third,ten; the rest of the balls are black. ...
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### A possible incorrect application of Law of Large numbers

A friend left this teaser for me. He asked me to first compute: $$\lim_{n \to \infty} \frac{\binom{2n}{n}}{2^{2n}}$$ Using Stirling's approximation (and another method), I got the answer as $0$. ...
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### Ice cream combinatorics question

An ice cream shop sells ice creams in $5$ possible flavours: vanilla, chocolate, strawberry, mango and pineapple. How many combinations of $3$ scoops cone are possible? [note: repetition of flavours ...
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### If bridges between islands collapse independently with probability $p$, what is the probability that islands remain connected?

This is a follow-up to Probability Question: Bridge problem. There are $n$ islands in the ocean. Each island is linked by a single bridge to each other island. The probability of each bridge ...
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### Probability Question: Bridge problem

There are $n$ islands in the ocean. Each island is linked by a single bridge between each and every unique pair of islands to ensure no island is isolated from the others. The probability of each ...
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### Probability formula related to distribution balls in boxes.

Problem Suppose there is a distribution of $N$ distinct balls in $n$ different boxes such that each ball has the same probability to be in any box. Let $A_i=\{\text{the i-th box is not empty}\}$. ...
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### Probability problem: n different balls in n different boxes

Problem Suppose $n$ different balls are distributed in $n$ different boxes. Calculate the probability that each box is not empty when distributed the balls. I'll define the sample space as ...
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### distribution of books among students

There are $p$ students and $q$ books where $q>p$ and all books are different, but each student will get a minimum of $1$ book and a maximum of $(p – 1)$ books. Find the total number of ways of ...
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### Subset Probability to Element Probability

Is there any way to match (or map) from Subset Propabilities to Element Probabilities? Suppose that John may select x-sized subsets from a population of N items. In every subset he has exactly x ...
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### Slots Machine Matching feature 2

I'm designing a slot machine. I need to find the number of combinations that two matching icons will appear side-by-side in a 3X5 window (3 rows, 5 reels (columns)) A match gives the user some ...
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### How to compute the expected number of unwatched rooms?

Imagine that I have a number of rooms, r, that I want to have watched. So I install one video camera in each room and have televisions that can show what's going ...
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### Probability of guessing a PIN-code

A friend and I recently talked about this problem: Say my friend feels a little adventurous and tells me that exactly three of four digits of his PIN-code are the same, what is the probability that I ...
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### A variation of the menage problem

A combinatorics problem I am chewing on without success is: 3 couples and 40 others are to be arranged randomly in a row. What is the probability that no two couples sit together ? I have looked at ...
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### A question involving the throw of seven dice, which of my answers is correct?

Seven dice are thrown, what is the probability that all numbers show up on the dice? My first answer uses the logic that if all numbers show up and you throw seven dice, then one number is repeated ...
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### Selecting 180 days from 366: the probability of even distribution across months, or not having September among the first 30

In a draft lottery containing the 366 days of the year (including February 29). Select 180 days (draw 180 without replacement). a) What is the probability that the 180 days drawn are evenly ...
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### Probability that a word contains at least 3 same consecutive letters?

Assume we have a word of length $n$ and an alphabet of length $26$ (the small letters a through z, if you want so. How likely is it that this word contains at least $k := 3$ consecutive letters of ...
Pick x random "birthdays", say $10^9$. What are the chance of a collision, given $2^{160}$ possible "days"? I'm trying to estimate the collision rate of sha1 hashes, but the calculation is too big ...