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I would make a distinction: for example in a queueing system the arrival times (or interval times) might be modelled by a Poisson process which would be time independent and would not be bound my initial conditions. This would be an example of a random process which outputs random variables. Service time in the queue would be dependent on the previous state(...

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This would be an interesting experiment in psychology or human behavior. What is the distribution of natural numbers "randomly" picked by volunteers? I suspect you'll see a bell-like distribution with a long tail where the 1st standard deviation is < 1000. Once you have that curve defined with experimental data, real probabilities can be calculated. ...

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First, I believe $P_{expected}=(\frac{1}{6})^2$ because it doesn't matter what the first die is, just that the other 2 match. The value you give for $P_{expected}$ is the value for a specific value, say, 6. There are six such values, so $(\frac{1}{6})^3•6=(\frac{1}{6})^2$.Second, there is a difference between theoretical probability and experimental ...

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What is not possible is to pick a random natural number in a way such that every natural number is equally likely. If you don't require every number to have the same probability, then one can certainly come up with a well-defined probabilistic experiment where every natural is a possible outcome -- for example, keep flipping a fair coin until you get tails, ...

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Consider this. If one small seed generates one long random sequence then it is impossible to create all possible sequences using a random function because for $n$ digit decimal seeds there can be only $10^n$ distinct possible sequences. If you have to create a sequence that doesn't belong to this set, you will have to use a different function... and in the ...

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Let $p_n$ be the $n$'th prime, and let $y_n = p_{n + p_{n}} \mod p_n$, i.e. $p_{n+p_{n}} = x_n p_n + y_n$ where $0 \le y_n < p_n$ and $x_n = \lfloor p_{n+p_n}/p_n \rfloor$. Now the point is that $r_n = p_{n+p_n}/p_n$ will tend to grow, but slowly: $p_n \sim n \log n$, $p_{n+p_n} \sim (n + p_n) \log(n + p_n) \sim n (\log n)^2$, so $r_n \sim \log n$. $... 0 As far as your second question goes, I don't believe prime numbers are randomly distributed. For example, for primes greater than 2, a prime number can only be odd. Further, since every 3rd odd number is divisible by 3, the only potential primes are of the form$6k±1$, for$k\in\mathbb N$(for primes greater than 3). Perhaps prime numbers are randomly ... 1 Using How to crack a Linear Congruential Generator, we can possibly crack it with the information given. We setup the following matrix using the four given values: $$A = \begin{pmatrix} 5 & 3 & 1 \\ 3 & 16 & 1 \\ 16 & 2 & 1 \\ \end{pmatrix}$$ The determinant of$A$yields$-141$, which has a prime factorization of$-141 = -1^1 \...

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Suppose the integers in the array are distinct. Therefore, only one permutation of the integers corresponds to a sorted array. Moreover, suppose you use the Fisher-Yates shuffle (or anything where each permutation is equally likely). Since there are $N!$ permutations, each permutation has probability $1/N!$ of appearing. Now, the expected number of trials ...

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