163
Chapter 11
The Law of Large Numbers
Let us move further away from frequency distribution and look at probability dis-
tributions. e frequency distribution that we have seen in Chapter 10 is an empiri-
cal pattern; what we are now going to see in this chapter and the rest of the book
are mathematical expressions.
The mathematical sciences particularly exhibit order, symmetry,
and limitation; and these are the greatest forms of the beautiful.
Aristotle
BOX 11.1 BIRTH OF PROBABILITY
Before the middle of the 17th century, the term probable meant approvable
and was applied in that sense, univocally, to opinion and to action. A prob-
able action or opinion was one such as sensible people would undertake or
hold in the circumstances. However, the term probable could also apply to
propositions for which there was good evidence, especially in legal contexts.
In the Renaissance times, betting was discussed in terms of odds such as “ten
to one,” and maritime insurance premiums were estimated based on intuitive
risks. However, there was no theory on how to calculate such odds or premiums.
e mathematical methods of probability arose in the correspondence of
Pierre de Fermat and Blaise Pascal (1654) on such questions as the fair divi-
sion of the stake in an interrupted game of chance.
164 ◾ Simple Statistical Methods for Software Engineering
Life Is a Random Variable
Results, in general, are random in nature; some could be in our favor and some not.
Process results do not precisely remain favorable all the time, neither do they become
unfavorable all the time. Results toggle between favor and disfavor, randomly.
The measure of the probability of an event is the ratio of the num-
ber of cases favorable to that event, to the total number of cases.
René Descartes
e discovery of probability goes back to the Renaissance times (see Box 11.1).
A process that toggles between favor and disfavor is called the Bernoulli process,
named after the inventor. Mathematically, a Bernoulli process takes randomly only
two values, 1 and 0. Repeated flipping a coin is a Bernoulli process; we get a head or
tail, success or failure, “1 or 0.” Every toss is a Bernoulli experiment. e Bernoulli
random variable was invented by Jacob Bernoulli, a Swiss mathematician (see Box
11.2 for a short biography).
Results from trials converge to the “expected value” as the number increases.
In an unbiased coin, the “expected value” of the probability of success (probability
of appearance of heads) is 0.5. More number of trials are closer to the value of the
probability of success. is is known as the law of large numbers. Using this law,
we can predict a stable long-term behavior. It took Bernoulli more than 20 years
to develop a sufficiently rigorous mathematical proof. He named this his golden
Fermat and Pascal helped lay the fundamental groundwork for the theory
of probability. From this brief but productive collaboration on the problem of
points, they are now regarded as joint founders of probability theory. Fermat
is credited with carrying out the first ever rigorous probability calculation. In
it, he was asked by a professional gambler why if he bet on rolling at least one
six in four throws of a die he won in the long term, whereas betting on throw-
ing at least one double-six in 24 throws of two dice resulted in his losing.
Fermat subsequently proved why this was the case mathematically. (http://
en.wikipedia.org/wiki/Problem_of_points)
Christiaan Huygens (1657) gave a comprehensive treatment of the subject.
Jacob Bernoulli’s Ars Conjectandi (posthumous, 1713) put probability on
a sound mathematical footing, showing how to calculate a wide range of
complex probabilities.
The Law of Large Numbers ◾ 165
theorem, but it became generally known as Bernoulli’s theorem. is theorem was
applied to predict how much one would expect to win playing various games of
chance.
From sufficient data from real-life events, we can arrive at a probability of suc-
cess (p) and trust that the future can be predicted based on this.
Prediction means estimation of two values: mean and variance (which denote
central tendency and dispersion).
In this chapter, we consider four distributions to describe four different ways of
describing the dispersion pattern.
1. Binomial distribution
e probability of getting exactly k successes in n trials is given by the fol-
lowing binomial expression:
P X k C p p
k
n k n k
( ) ( )= = −
−
1 (11.1)
where n is the number of trials, p is the probability of success (same for each
trial), and k is the number of successes observed in n trials, calculated as
follows:
Mean = np (11.2)
Variance = np(1 − p) (11.3)
Equation 11.1 is a paradigm for a wide range of contexts. In service man-
agement, success is replaced by arrival, and the Bernoulli process is called
arrival-type process. In software development processes, we prefer to use the
term success. e coefficient C is a binomial coefficient, hence the name bino-
mial distribution.
Software development processes may consist of two components:
a. An inherent Bernoulli component that complies with the law of large
numbers
b. Influences from spurious noise factors
Bernoulli distribution is used in statistical process control. e spurious
noise factors must be identified, analyzed, and eliminated. For example, in
service-level agreement (SLA) compliance data, one may find both these
components. If the process is restricted to the Bernoulli type, the process
is said to be under statistical control. (Shewhart called this variation due to
“common causes” and ascribed spurious influences to special causes.)
Equation 11.1 is used in the quality control of discrete events.
166 ◾ Simple Statistical Methods for Software Engineering
Example 11.1: Binomial Distribution of SLA Compliance
QUESTION
From the previous year’s deliveries, it has been estimated that the probability of
meeting SLA in an enhancement project is 90%. Find out the probability of meet-
ing SLA in at least 10 of 120 deliveries scheduled in the current year. Plot the
related binomial distribution.
BOX 11.2 JACOB BERNOULLI 16541705
Nature always tends to act in the simplest way.
Jacob Bernoulli
Jacob Bernoulli gave a mathematical footing to the theory of probability. e
term Bernoulli process is named after him. A well-known name in the world
of mathematics, the Bernoulli family has been known for their advancement
of mathematics. Originally from the Netherlands, Nicolaus Bernoulli, Jacob’s
father, moved his spice business to Basel, Switzerland. Jacob graduated from
the University of Basel with a master’s degree in philosophy in 1671 and a
master’s degree in theology in 1676. When he was working toward his mas-
ter’s degrees, he would also study mathematics and astronomy. In 1681, Jacob
Bernoulli met mathematician Hudde. Bernoulli continued to study math-
ematics and met world-renowned mathematicians such as Boyleand Hooke.
Jacob Bernoulli saw the power of calculus and is known as one of the
fathers of calculus. He also wrote a book called Ars Conjectandi, published in
1713 (8 years after his death).
Bernoulli added upon Cardano’s idea of the law of large numbers. He
asserted that if a repeatable experiment had a theoretical probability p of turn-
ing out in a certain “favorable” way, then for any specified margin of error, the
ratio of favorable to total outcomes of some (large) number of repeated trials of
that experiment would be within that margin of error. By this principle, obser-
vational data can be used to estimate the probability of events in real-world
situations. is is what is now known as the law of large numbers. Interestingly,
when he wrote the book, he named this idea the “Golden theorem.”
Bernoulli had received several awards and/or honors. One of the honors
given to him was a lunar crater named after him. In Paris, there is a street
named after the Bernoulli family. e street is called Rue Bernoulli.
The Law of Large Numbers ◾ 167
ANSWER
You can solve Equation 11.1 by keeping p = 0.9, n = 120, and k = 10. is will
give the probability of exactly 10 deliveries meeting the SLA. We are assessing the
chance of getting exactly 10 successes.
Alternatively, use MS Excel function BINOM.DIST.
e answer is zero. e number is too small, 4.04705E-97.
Figure 11.1 shows the binomial probability distribution of SLA compliance.
is may be taken as a process model. One can notice the upper and lower bound-
aries of the density function, approximately from 91 to 118 trials; one can also note
the central tendency, which is exactly the mean.
BOX 11.3 TESTING RELIABILITY USING
NEGATIVE BINOMIAL DISTRIBUTION
To test reliability, we randomly selected and run test cases covering usage.
Executing a complete test library is costly, so we resort to sampling. We can
choose inverse sampling and choose and execute test cases randomly until a
preset number of defects are found (unacceptable defect level). If this level is
reached, the software is rejected. Using regular sampling and under binomial
distribution, we can do an acceptance test, but we might require to execute
a significantly large number of test cases to arrive at an equivalent decision.
Inverse sampling under NBD is more efficient in user acceptance testing.
With this information, we can construct the negative binomial distribu-
tion of defects. e salient overall point of the comparison is that, unless the
software is nearly perfect, the negative binomial mode of sampling brings
about large reductions in the average number of executions over the binomial
mode of sampling for identical false rejection and false acceptance risks [1].
0
0.02
0.04
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0.08
0.10
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42
46
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58
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118
k
Figure 11.1 Binomial probability of SLA compliance (p = 0.9, n = 120).
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