121
Chapter 12: Probability
Problem numbers in italics indicate that the solution is included in the
Student’s 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 4ball set, the number of
possible configurations is:
C
4,2
()
=
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
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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.
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 Spring '03
 Nefzi
 Physical chemistry, Probability, pH, Probability distribution, Probability theory, Probability space

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