Monte Carlo methods keep track of the colliding particle’s path after the collision and determine the probability of the production of pions. The particle’s path after the collision determines whether or not a new particle, a pion, is created. When a particle collides with a nucleus, a nuclear cascade is produced. In the 1950s, Monte Carlo methods were used to study nuclear cascades.
Monte Carlo methods are often used in Physics to determine neutron trajectories or to simulate atomic clusters. More examples on the Monte Carlo algorithm can be found here. As I have mentioned ealier, by the Law of Large Numbers, if you increase the number of simulated tournaments, you will notice that either player will always be the winner because the expected result will always come closer because of the large number of simulations. There are some instances where it is the other way around. We can also compute the mean win for all the 10 tournaments they have played. Assume they each have played 10 tournaments and we have gotten the minimum and maximum number of wins (assuming winning the tournament has the same number of wins as all the other tournaments). Here is an example where we have two players, each with varying skills and talent. The flow chart of the process goes like this: There are many ways of approaching a Monte Carlo analysis, but a good starting point is a flow chart of the processing being modeled.įor the purposes of this simulation, I will be focus on a tennis tournament simulation based on game winnings. So, a Monte Carlo analysis should be preceded by a sensitivity analysis to determine what the important parameters are. Typically, in any activity, there are a few critical inputs, and it is these which should be the focus of the analysis, not trying to attribute complex distributions to minor variables. Whilst Monte Carlo methods can be a valuable aid to decision making, they should not be used without an understanding of the process being modeled. But, if the most likely outcome is also known, then the outcome can be simulated by a Triangular distribution. Generally, when not much is known about the distribution of an outcome (say, only its smallest and largest values), it is possible to use uniform distribution. Triangular distribution is therefore often used in business decision making, particularly in simulations. In the sample program, I will be using a Triangular distribution, which is is a continuous probability distribution with a lower limit a, mode c, and upper limit b. Normal distribution, continuous uniform distribution, beta distribution, and Gamma distribution are well known absolutely continuous distributions. If the distribution of X is continuous, then X is called a continuous random variable.
That is, the probability that X attains the value a is zero, for any number a. That is equivalent to saying that for random variables X with the distribution in question, Pr = 0 for all real numbers a. Probability distribution identifies either the probability of each value of an unidentified random variable (when the variable is discrete), or the probability of the value falling within a particular interval (when the variable is continuous). There are many different numerical experiments that can be done, probability distribution is one of them. Other performance or statistical outputs are indirect methods which depend on the applications. The direct output of the Monte Carlo simulation method is the generation of random sampling. Monte Carlo methods have three characteristics: It says that if you generate a large number of samples, eventually, you will get the approximate desired distribution. Monte Carlo algorithms work based on the Law of Large Numbers. Monte Carlo methods tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithm. Because of their reliance on repeated computation and random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. Monte Carlo methods are often used when simulating physical and mathematical systems. A class of computational algorithms that rely on repeated random sampling to compute their results are called the Monte Carlo methods.
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