Notes

Monte-Carlo
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In application to space and oil exploration problems, Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods
Monte Carlo methods vary, but tend to follow a particular pattern:
Define a domain of possible inputs.
Generate inputs randomly from a probability distribution over the domain.
Perform a deterministic computation on the inputs.
Aggregate the results.
Uses of Monte Carlo methods require large amounts of random numbers, and it was their use that spurred the development of pseudorandom number generators, which were far quicker to use than the tables of random numbers that had been previously used for statistical sampling.

Definition
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Distinction between a simulation, a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to determine the properties of some phenomenon (or behavior). Examples:
Simulation: Drawing one pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin: If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.
Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.
Monte Carlo simulation: Drawing a large number of pseudo-random uniform variables from the interval [0,1], and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a Monte Carlo simulation of the behavior of repeatedly tossing a coin.

How random
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Monte Carlo simulation methods do not always require truly random numbers to be useful — while for some applications, such as primality testing, unpredictability is vital.[10] Many of the most useful techniques use deterministic, pseudorandom sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good simulations is for the pseudo-random sequence to appear "random enough" in a certain sense.

Pseudo-random number sampling algorithms are used to transform uniformly distributed pseudo-random numbers into numbers that are distributed according to a given probability distribution.
Monte Carlo simulation versus "what if" scenarios

What-the-if
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There are ways of using probabilities that are definitely not Monte Carlo simulations — for example, deterministic modeling using single-point estimates. Each uncertain variable within a model is assigned a “best guess” estimate. Scenarios (such as best, worst, or most likely case) for each input variable are chosen and the results recorded.[11]
By contrast, Monte Carlo simulations sample probability distribution for each variable to produce hundreds or thousands of possible outcomes. The results are analyzed to get probabilities of different outcomes occurring.[12] For example, a comparison of a spreadsheet cost construction model run using traditional “what if” scenarios, and then run again with Monte Carlo simulation and Triangular probability distributions shows that the Monte Carlo analysis has a narrower range than the “what if” analysis.[examples needed] This is because the “what if” analysis gives equal weight to all scenarios (see quantifying uncertainty in corporate finance), while Monte Carlo method hardly samples in the very low probability regions. The samples in such regions are called "rare events

====>> Primality Test -- Look in detail
All primes are of the form 6k ± 1, with the exception of 2 and 3. This is because all integers can be expressed as (6k + i) for some integer k and for i = −1, 0, 1, 2, 3, or 4; 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k + 3). So a more efficient method is to test if n is divisible by 2 or 3, then to check through all the numbers of form 6k ± 1 <=sqrt n. This is 3 times as fast as testing all m.

public static boolean isPrime(long n) {
if (n <= 3) {
return n > 1;
} else if (n % 2 == 0 || n % 3 == 0) {
return false;
} else {
for (int i = 5; i * i <= n; i += 6) {
if (n % i == 0 || n % (i + 2) == 0) {
return false;
}
}
return true;
}
}
Note - In java div and rem ops can be optimized by using the bit counterparts
Note - We are avoiding sqrt by trading it with square up operation and comparing with original operand

Generalising further, it can be seen that all primes are of the form c#k + i for i < c# where i represents the numbers that are coprime to c# and where c and k are integers. For example, let c = 6. Then c# = 2 * 3 * 5 = 30. All integers are of the form 30k + i for i = 0, 1, 2,...,29 and k an integer. However, 2 divides 0, 2, 4,...,28 and 3 divides 0, 3, 6,...,27 and 5 divides 0, 5, 10,...,25. So all prime numbers are of the form 30k + i for i = 1, 7, 11, 13, 17, 19, 23, 29 (i.e. for i < 30 such that gcd(i,30) = 1). Note that if i and 30 are not coprime, then 30k + i is divisible by a prime divisor of 30, namely 2, 3 or 5, and is therefore not prime.
As c → ∞, the number of values that c#k + i can take over a certain range decreases, and so the time to test n decreases.
For this method, it is also necessary to check for divisibility by all primes that are less than c. Observations analogous to the preceding can be applied recursively, giving the Sieve of Eratosthenes.

===> Note - Factorization is not computer friendly op -- Why?? .. How to avoid ???
====> Graph of logN (base 10 and base e)
====> How to calculate ratio of Pi and e