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Section 12.1
The Nature of
Probability
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
History
The Nature of Probability
Empirical Probability
The Law of Large Numbers
12.1-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
History
Probability is used in many areas,
including public finance, medicine,
insurance, elections, manufacturing,
educational tests and measurements,
genetics, weather forecasting,
investments, opinion polls, the natural
sciences, and games of chance.
12.1-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
History
Mathematical problems relating to
games of chance were studied by a
number of mathematicians of the
Renaissance:
Italy’s Girolamo Cardano (1501–1576)
France’s Blaise Pascal (1623–1662)
and Pierre de Fermat (1601–1665)
Dutch Christian Huygens (1629–1695)
Swiss Jacob Bernoulli (1654–1705)
Pierre-Simon de Laplace (1749–1827)
12.1-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Definitions
An experiment is a controlled
operation that yields a set of results.
The possible results of an experiment
are called its outcomes.
An event is a subcollection of the
outcomes of an experiment.
12.1-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Definitions
Empirical probability is the relative
frequency of occurrence of an event
and is determined by actual
observations of an experiment.
Theoretical probability is
determined through a study of the
possible outcomes that can occur for
the given experiment.
12.1-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Empirical Probability
(Relative Frequency
The empirical probability of an event,
P(E), can be determined by the
following formula.
number of times
event E has occurred
P(E) 
total number of times the
experiment has been performed
12.1-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Heads Up!
In 100 tosses of a fair coin, 44 landed
heads up. Determine the empirical
probability of the coin landing heads
up.
Solution
Let E be the event that the coin lands
heads up.
44
P(E) 
 0.44
100
12.1-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Empirical Probability in Genetics
Using empirical probability, Gregor
Mendel (1822–1884) developed the
laws of heredity by crossbreeding
different types of “pure” pea plants
and observing the relative frequencies
of the resulting offspring. These laws
became the foundation for the study of
genetics.
12.1-9
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Empirical Probability in Genetics
12.110
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Empirical Probability in Genetics
Mendel called traits such as yellow
color and round seeds dominant
because they overcame or “dominated”
the other trait. He labeled the green
color and the wrinkled traits
recessive.
12.111
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Empirical Probability in Genetics
12.112
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Empirical Probability in Genetics
12.113
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Empirical Probability in Genetics
Mendel concluded that the sex cells
(now called gametes) of the pure
yellow (dominant) pea plant carried
some factor that caused the off-spring
to be yellow and that the gametes of
the green variety had a variant factor
that “induced the development of
green plants.” In 1909, Danish
geneticist W. Johannsen called these
factors “genes.”
12.114
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
The Law of Large Numbers
The law of large numbers states that
probability statements apply in practice
to a large number of trials, not to a
single trial. It is the relative frequency
over the long run that is accurately
predictable, not individual events or
precise totals.
12.115
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Tossing a Fair Coin
12.116
Copyright 2013, 2010, 2007, Pearson, Education, Inc.