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Unformatted text preview: Chapter 12: Probability Problem numbers in italics indicate that the solution is included in the Students Solutions Manual. Questions on Concepts Q12.1) What is the difference between a configuration and a permutation? A configuration is an unordered arrangement of objects. A permutation is a specific order of an arrangement of objects. Q12.2) What are the elements of a probability model, and how do they differ for continuous and discrete variables? A probability model consists of a sample space containing the possible values for a variable, and the corresponding probabilities that the variable will assume a value in the sample space. In the discrete case, the sample space consists of a set of specific values a variable can assume, where in the continuous case there is a range of values the variable can assume. Q12.3) How does Figure 12.2 change if one is concerned with two versus three colored- ball configurations and permutations? For the case where two balls are chosen from the 4-ball set, the number of possible configurations is: C 4,2 ( 29 = 4! 2!2! = 6 Therefore in Figure 12.2, there will be six rows in the left column corresponding to the 6 possible configurations, with each configuration having two associated permutations. Q12.4) What must the outcome of a binomial experiment be if P E = 1? If the probability of a successful trial is unity, the probability of observing j successful trials out of n total trials is unity. That is, every trial will be successful. Q12.5) Why is normalization of a probability distribution important? What would one have to consider when working with a probability distribution that was not normalized? A variable will always assume some value from the sample set; therefore, normalization of a probability distribution ensures that the sum of probabilities for the variable assuming values contained in the sample set is equal to unity. If the 12-1 Chapter 12/Probability probability distribution is not normalized, then each individual probability should be divided by the sum of all probabilities. Q12.6) What properties of atomic and molecular systems could you imagine describing using probability distributions? Electron orbital densities, distributions of bond lengths or molecular geometries, locations of particles in space, etc. In quantum mechanics, the square modulus of the wavefunction is simply a statement of a probability distribution. Q12.7) When is the higher moment of a probability distribution more useful as a benchmark value as opposed to simply using the mean of the distribution? When the spread or width of the distribution is of interest in addition to the average value. Particle velocity distributions serve as a good example of this issue. Consider motion in a single dimension. Since particles are just as likely to be moving in the positive and negative direction, the average velocity is equal to zero (see Problem P12.23); however, the width of the velocity distribution will be finite, as can be judged by considering higher moments of the distribution.finite, as can be judged by considering higher moments of the distribution....
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- Fall '09