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72 Modern Analytical Chemistry
Binomial Distribution The binomial distribution describes a population in which
binomial distribution
Probability distribution showing chance the values are the number of times a particular outcome occurs during a fixed num-
of obtaining one of two specific ber of trials. Mathematically, the binomial distribution is given as
outcomes in a fixed number of trials.
N!
-
(
PX, N) = ´ p X ( ´1 - p) N X
XN - X)!
!(
where P(X,N) is the probability that a given outcome will occur X times during N
trials, and p is the probability that the outcome will occur in a single trial.* If you
flip a coin five times, P(2,5) gives the probability that two of the five trials will turn
up “heads.”
A binomial distribution has well-defined measures of central tendency and
spread. The true mean value, for example, is given as
m= Np
and the true spread is given by the variance
2
s = Np(1 – p)
or the standard deviation
s= Np(1 -p)
The binomial distribution describes a population whose members have only
certain, discrete values. A good example of a population obeying the binomial dis-
homogeneous tribution is the sampling of homogeneous materials. As shown in Example 4.10, the
Uniform in composition. binomial distribution can be used to calculate the probability of finding a particular
isotope in a molecule.
4
EXAMPLE .10
12
13
Carbon has two common isotopes, C and C, with relative isotopic
abundances of, respectively, 98.89% and 1.11%. (a) What are the mean and
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standard deviation for the number of C atoms in a molecule of cholesterol?
(b) What is the probability of finding a molecule of cholesterol (C 27 H 44 O)
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containing no atoms of C?
SOLUTION
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The probability of finding an atom of C in cholesterol follows a binomial
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distribution, where X is the sought for frequency of occurrence of C atoms, N
is the number of C atoms in a molecule of cholesterol, and p is the probability
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of finding an atom of C.
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(a) The mean number of C atoms in a molecule of cholesterol is
m= Np =27 ´0.0111 = 0.300
with a standard deviation of
s= ( )( .0111 )(1 - .0111 ) =.172
0
0
0
27
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(b) Since the mean is less than one atom of C per molecule, most
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molecules of cholesterol will not have any C. To calculate
*N! is read as N-factorial and is the product N ´(N –1) ´(N –2) ´ ... ´1. For example, 4! is 4 ´3 ´2 ´1, or 24.
Your calculator probably has a key for calculating factorials.
