Permutations, Combinations, and Probability

Each separate arrangement of all or a part of a set of things is called a permutation. The number of permutations of n things taken r at a time, written 

If an event can occur in p ways and fail to occur in q ways, all ways being equally likely, the probability of its occurrence is p/(p + q), and that of its failure q/(p + q). 

Linear Equations A linear equation is one of the first degree (i.e., only the first powers of the variables are involved), and the process of obtaining definite values for the unknown is called solving the equation. Every linear equation in one variable is written Ax + B = 0 or x = —B/A. Linear equations in n variables have the form 

The solution of the system may then be found by elimination or matrix methods if a solution exists (see Matrix Algebra and Matrix Computations )- 

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As is well known, if D(z) is a periodic function with period d, then its Fourier transform is a periodic set of delta functions of period 1/d and the intensity distribution in reciprocal space consists of discrete maxima also spaced 1/d apart. If the distribution function is not periodic but consists of two randomly arrayed periods, then it is necessary to consider all possible combinations and permutations, suitably weighted by their probability of occurrence. 

PERMUTATIONS AND COMBINATIONS PROBABILITY DENSITY FUNCTION PROBABLE ERROR NORMAL ERROR CURVE STATISTICS (A Primer)... 

With these specifications, and with the appropriate neutral particle-plasma collision terms put into the combined set of neutral and plasma equations, internal consistency within the system of equations is achieved. Overall particle, momentum and energy conservation properties in the combined model result from the symmetry properties of the transition probabilities W indices of pre-collision states may be permuted, as well as indices of postcollision states. For elastic collisions even pre- and post collision states may be exchanged in W. 

For each permutation, the phases of all other reflections are generated using Sayre equations. Thus, direct methods always result in more than one array of phases, and the problem is reduced to selecting the correct solution, if one exists. Several different figures merit and/or their combinations have been developed and are used to evaluate the probability and the relationships between phases. Thus, the solutions are sorted according to their probability - from the highest to the lowest. Then each solution is analyzed and evaluated starting from the one that is most probable. 

It is possible to recombine any number of parent genes with the available methods, which raises the question of what is the optimal number. Similar to determining the optimal mutation rate for random mutagenesis, the answer will depend on the number of screened mutants and the additivity of the combined mutations. It could be advantageous to screen all the permutations of mutations from the parents. Assuming independent and additive recombination, the probability Pd that an offspring has d mutations is given by... 

The expression (Equation 13) approximates the probabilities (Equation 6) and (Equation 7) by letting the coalescence probability of a range of bubble sizes depend on the number available in that range while ignoring all the detailed permutations of different size combinations. For sufficiently large N and k, this is a satisfactory approximation. 

Next consider the various probabilities when five coins are tossed simultaneously. Table 14.2 gives the results with their probabilities. The probability of obtaining three tails and two heads or three heads and two tails is 5/16 = 0.313. The laborious work of solving such problems can be greatly simplified by introducing mathematical conventions called permutations and combinations. 

Lloyd and Tye (1995, p. 42) advise us therefore to be very careful in calculating probabilities of combined occurrences to be sure we have properly counted the various possible permutations and combinations. 

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