# Monte Carlo method

Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results; typically one runs simulations many times over in order to obtain the distribution of an unknown probabilistic entity. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to obtain a closed-form expression, or infeasible to apply a deterministic algorithm. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration and generation of draws from a probability distribution. The modern version of the Monte Carlo method was invented in the late 1940s by Stanislaw Ulam, while he was working on nuclear weapons projects at the Los Alamos National Laboratory. Introduction Monte Carlo method applied to approximating the value of π. Monte Carlo methods vary, but tend to follow a particular pattern: History Definitions Related:  mathematics

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Dr Hannah Fry: the mathematical models that underpin our sexual success | Science What are the odds? Or how mathematician Peter Backus weighed up his chances of finding love… Just as it’s not possible to calculate precisely how many alien life forms there are, it’s also not possible to calculate exactly how many potential partners you may have. It also applies to maths student Peter Backus’s well-publicised quest to see whether there were intelligent, socially advanced women of the same species out there for him to date. 1. 2. 3. 4. 5. 6. 7. Leaving him with just 26 women in the whole world he would be willing to date. Personally, I think that he is being a little picky. I think there’s room to be a bit more generous. 1. 5. 6. 7. Almost a thousand potential partners across a city, then. But there is another issue. Strangely though, opening our minds to all potential partners seems to be the opposite of what we do when we’re single. The statistics of sex In 1996 Swedish scientists asked 2,810 people how many sexual partners they’d had. The happily ever after

Chapter 12: Classes and objects Warning: the HTML version of this document is generated from Latex and may contain translation errors. In particular, some mathematical expressions are not translated correctly. 12.1 User-defined compound types Having used some of Python's built-in types, we are ready to create a user-defined type: the Point. Consider the concept of a mathematical point. A natural way to represent a point in Python is with two floating-point values. An alternative is to define a new user-defined compound type, also called a class. A class definition looks like this: class Point: pass Class definitions can appear anywhere in a program, but they are usually near the beginning (after the import statements). This definition creates a new class called Point. By creating the Point class, we created a new type, also called Point. blank = Point() The variable blank is assigned a reference to a new Point object. 12.2 Attributes We can add new data to an instance using dot notation: >>> blank.x = 3.0 >>> blank.y = 4.0

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Ding Dong Bell The sound of bells In East Anglia, as you look across the fens, villages appear almost like little islands (indeed some of them were islands before the fens were drained) and these villages are dominated by big churches with tall towers. In the past people regulated their lives and passed messages by ringing church bells, which could be heard for miles around, telling the time of day, and giving news of births, marriages and deaths in a parish. The following quotation comes from the ringer's rules from Southhill in Bedfordshire "When mirth and pleasure is on the wing we ring; at the departure of a soul we toll". Bell ringing is good exercise for the body and mind, the bells are heavy and the bellringers have to remember the changes. The mathematics of the changes With four bells there are of course many more possibilities; there are twenty four different permutations or orders in which the bells can be rung and there are four bell ringing changes. Find the vertex labelled 1234.

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