Games of chance

Monte Carlo search methods are stochastic techniques based on the use of random numbers and probability statistics to sample conformational space. The name Monte Carlo was originally coined by Metropolis and Ulam [4] during the Manhattan Project of World War II because of the similarity of this simulation technique to games of chance. Today a variety of Monte Carlo (MC) simulation methods are routinely used in diverse fields such as atmospheric studies, nuclear physics, traffic flow, and, of course, biochemistry and biophysics. In this section we focus on the application of the Monte Carlo method for... 

Almost everybody is familiar with the notion of probability as applied to games of chance. When discussing dice games, we assign the probability 1/6 to the event a one comes up because there are six faces of the die that are equally likely to come up and the desired face is, therefore, only one possibility out of six. Operationally, the number 1/6 is taken to mean that, in a long sequence of N tosses of the die, approximately 1/6 -N ones will show up. In other words, the... 

The symbol E[faX)] or E[] is referred to as the mathematical expectation of the function fa The use of this term stems from the early applications of the theory to games of chance, where it was used to denote the average amount a gambler could expect to win. The basic rules for manipulating the expectation symbol E are direct consequences of its definition, and are stated below for easy reference. 

The statistics of this process is identical to those pertaining to the tossing of a coin. The mathematics was first worked out with respect to games of chance by de Moivre, in 1733. It is formally described by the binomial distribution. 

The pioneers in mathematical statistics, such as Bernoulli, Poisson, and Laplace, had developed statistical and probability theory by the middle of the nineteenth century. Probably the first instance of applied statistics came in the application of probability theory to games of chance. Even today, probability theorists frequently choose... 

Medical genetics. 2. Science—Public opinion. 3. Cats. 4. Games of chance (Mathematics) 5. Hunting. 6. Manned space flight. 7. Violence in mass media. I. Title. 

For some variables, for example, the relative collision velocity, the cumulative distribution function does not have closed form, and then a third Monte Carlo method must be adopted. Here, another random number R is used to provide a value of v, but a decision on whether to accept this value is made on the outcome of a game of chance against a second random number. The probability that a value is accepted is proportional to the probability density in the statistical distribution at that value. The procedure is repeated until the game of chance is won, and the successful value of v is then incorporated into the set of starting parameters. 

Amato, I. (1992). Speeding up a chemical game of chance. Science 257, 330 331. 

Games of chance inspired several early investigations of the random sum representing the number of successes in n Bernoulli trials ... 

Bayesian procedures are important not only for estimating parameters and states, but also for decision making in various fields. Chapters 6 and 7 include applications to model discrimination and design of experiments further applications appear in Appendix C. The theorem also gives useful guidance in economic planning and in games of chance (Meeden, 1981). 

Four coins are provided for a game of chance. Three of the coins are fair, giving y(Heads) = p(Tails) = 1/2 for a fair toss, but the fourth coin has been altered so that both faces show Heads, giving p(Heads) = 1 and y(Tails) = 0. 

Everyone is familiar with the common sense concept of probability as a way to assess the likelihood of a desirable outcome in a game of chance. The purpose of this section is to give a brief introduction to probability in a form suited for scientific work. 

No alcohol and games of chance are accepted at trade display. 

Annealing is not the only process that may be used as a model for the development of probabilistic optimization algorithms. Bohachevsky, Johnson, and Stein [5] identified an analogy between stochastic optimization and a biased game of chance and used that analogy to estimate... 

As mentioned earlier, the limiting probabilities occur only in the ratio pjpm and the value of the denominator, Qc, is therefore not required. Hence, in the canonical ensemble, the acceptance of a trial move depends only on the Boltzmann factor of the energy difference, AU, between the states m and n. If the system looses energy, then the trial move is always accepted if the move goes uphill, that is, if the system gains energy, then we have to play a game of chance on exp(—fi AU) [6]. In practical terms, we have to calculate a random number, uniformly distributed in the interval [0,1]. If < exp(—p AU), then we accept the move, and reject otherwise. 

Schmid E F, Smith D A (2004). Is pharmaceutical R D just a game of chance or can strategy make a difference Drug Disc. Today, 9 18-26. 

The problem can be illustrated with respect to a simple game of chance which I call coins and which comes in two versions. In each version a fair coin is tossed three times and a Swiss franc is paid for each toss. A reward is paid if and only if all three tosses are successful. In the first version, the reward is Sfr 25.00 and in the second Sfr 20.00. However, for the first game the three francs must be paid up front. In the second they are only paid to see each toss and the player can stop any time he or she likes and start another game. Which of the two games is more valuable. ... 

Life is not a haphazard game of chance, but an unfoldment and development of its own powers manifesting in perfect Law. Let us, then, try to understand this Life which is Eternal Law, pervaded by an Intelligence with Order and Wisdom, and having understood, let us work for the more perfect unfoldment of our earth and the forces which lie beneath its surface for this Law applies to agriculture, to science, to the production of food, to the use of minerals and metals, to the building of cities, to the use of electricity and all natural forces. When man finally learns to use these forces, he will be able to press forward and onward to the final goal, which is the perfection of the earth and of his own species. 

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