All Questions
Tagged with integer-partitions probability
24
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Probability that the maximum number of dice with the same face is k
Let say we have $N$ dice with 6 faces. I'm asking my self, what is the probability that the maximum number of dice with the same face is $k$?
In more precise terms, what is the size of this set?
\...
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Closed form version of partition and combination problem
We have an original set of n integers, $0, ... , (n-1)$. From these integers generate a set of $k$-tuples, with repetition where order does not matter. From this, want to know the probability that any ...
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How to rigorously interpret and transform "equal chance" in different ways?
Put $100$ identical balls into $10$ identical boxes in a way that each ball enters each box with an equal chance. What's the probability that no box is empty?
I have solved it but like to discuss ...
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"On the Probability of Sequences in the Genoese Lottery" by Euler. How did he do it?
This a problem solved by Leonard Euler. Translated English version is available in Euler archives.[E338]On the Probability of Sequences in the Genoese Lottery Euler.
I have difficulty in understanding ...
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Combination with Restriction and Repetition
I have a number $x$, let's say $5$, and I want to sort the number out into $4$ digits so that the sum of the digits is equal to $5$, but the value of each digit cannot exceed $3$. $0$ would be an ...
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Distribution of a method for generating random whole numbers with a given sum
I am writing a computer program which involves generating 5 random non negative integer numbers with a specific sum, namely 30.
I've found this method, but I don't know if it really generates a ...
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A Question related to probability.
Question is described here last question G1†
Solution is here last solution
Please explain the solution in simple language.
And please explain how to do it using partition.
My approach using ...
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Number theoretic calculation of probability problem involving partitions.
This question asks for a specific miscalculation of how large chance that I get at least one call every day of the week if I in total get 12 calls during a week.
If we instead consider the question ...
- 26.1k
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Add together numbers to make final number?
I have this problem where I want to add some numbers together to make a final number. I ask how many ways are there to make this final number using exactly k numbers? Here is the condition:
(1) The ...
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1
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Uniqueness of a P-measurable random variable
"Let $X$ be a discrete random variable on a discrete sample space. Let P be a partition of that sample space.
a) Show that $E[X|P]$ is the unique P-measurable random variable $Y$, ie $Y$ = $E[X|P]$, ...
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What form does the Law of Total Probability take if the partition you use is generated by the random variable Y?
The Law of Iterated Expection Looks like E{E(X|Y)} when the partition you use is generated by the random variable Y rather than Ω. What happens when you use such a partition on the Law of Total ...
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How to calculate the distribution of the last digit of an integer partition?
Definition
Let's denote $P(n,k)$ represents the set of ordered partitions of integer $n$ into $k$ parts. For example, $P(5,3)=\{(1,1,3),(1,2,2),(1,3,1),(2,1,2),(2,2,1),(3,1,1)\}$
(there is another ...
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Partitions and Truncations
Let $P(x)=\sum_{n=0}^{\infty} p_nx^n$ be the partition generating function, and let $P^*(x)=\sum_{n=0}^{\infty} p^*_nx^n$, where
$$p^*_n = \binom{\text{number of partitions of }n}{\text{into an even ...
- 1,283
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Compositions and Partitions
For a nonnegative integer $n$, a composition of $n$ means a partition in which the order of the parts matters. For example, the compositions of $3$ are $3$, $2+1$, $1+2$, and $1+1+1$.
Consider the ...
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Partition Theorem and Markov Chains
Suppose a Markov chain has $s$ states, $S = {1, 2, . . . , s}$, with PTM $P =$ ($p_{ij}$). That is, $p_{ij} = P[X_{n+1} = j | X_n = i]$. Use the Partition Theorem to verify that if $X_n ∼ ν$, then $X_{...
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