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4.4 THE POISSON DISTRIBUTION 111
when n is large and p is small. In this case, the Poisson distribution can be used to
approximate the binomial distribution. In other words,
where l = np.
To illustrate the use of the Poisson distribution for computing probabilities, let us
consider the following examples.
EXAMPLE 4.4.1
nC x p x q n - x « e-l l x
, x = 0, 1, 2, . . .
x!
In a study of drug-induced anaphylaxis among patients taking rocuronium bromide as
part of their anesthesia, Laake and Røttingen (A-7) found that the occurrence of anaphylaxis
followed a Poisson model with l = 12 incidents per year in Norway. Find the probability
that in the next year, among patients receiving rocuronium, exactly three will
experience anaphylaxis.
Solution:
By Equation 4.4.1, we find the answer to be
P1X = 32 = e-12 12 3
3!
= .00177
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EXAMPLE 4.4.2
Refer to Example 4.4.1. What is the probability that at least three patients in the next
year will experience anaphylaxis if rocuronium is administered with anesthesia
Solution:
We can use the concept of complementary events in this case. Since
P1X … 22 is the complement of P1X Ú 32, we have
P1X Ú 32 = 1 - P1X … 22 = 1 - 3P1X = 02 + P1X = 12 + P1X = 224
= 1 - c e-12 12 0
0!
= 1 - 3.00000614 + .00007373 + .000442384
= 1 - .00052225
= .99947775
+ e-12 12 1
1!
+ e-12 12 2
d
2!
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In the foregoing examples the probabilities were evaluated directly from the equation.
We may, however, use Appendix Table C, which gives cumulative probabilities for
various values of l and X.
EXAMPLE 4.4.3
In the study of a certain aquatic organism, a large number of samples were taken from a
pond, and the number of organisms in each sample was counted. The average number