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Unformatted text preview: es of chance. The importance of the theory grew rapidly,
particularly in the 20th century, and it now plays a central role in risk assessment, statistics,
data networks, operations research, information theory, control theory, theoretical computer
science, quantum theory, game theory, neurophysiology, and many other ﬁelds.
The core concept in probability theory is that of a probability model. Given the extent of
the theory, both in mathematics and in applications, the simplicity of probability models
is surprising. The ﬁrst component of a probability model is a sample space, which is a set
whose elements are called outcomes or sample points. Probability models are particularly
simple in the special case where the sample space is ﬁnite,2 and we consider only this
case in the remainder of this section. The second component of a probability model is a
class of events, which can be considered for now simply as the class of all subsets of the
sample space. The third component is a probability measure, which can be regarded here as
the assignment of a nonnegative number to each outcome, with the restriction that these
numbers must sum to one over the sample space. The probability of an event is the sum of
the probabilities of the outcomes comprising that event.
These probability models play a dual role. In the ﬁrst, the many known results about various
classes of models, and the many known relationships between models, constitute the essence
of probability theory. Thus one often studies a model not because of any relationship to the
real world, but simply because the model provides an example or a building block useful
for the theory and thus ultimately for other models. In the other role, when probability
theory is applied to some game, experiment, or some other situation involving randomness,
a probability model is used to represent a trial of the experiment (in what follows, we refer
to all of these random situations as experiments).
For example, the standard probability model for rolling a die uses {1, 2, 3, 4, 5, 6} as the
sample space, with each possible outcome having probability 1/6. An odd result, i.e., the
subset {1, 3, 5}, is an example of an event in this sample space, and this event has probability
1/2. The correspondence between model and actual experiment seems straightforward here.
The set of outcomes is the same and, given the symmetry between faces of the die, the choice
1
It would be appealing to show how probability theory evolved from realworld situations involving
randomness, but we shall ﬁnd that it is diﬃcult enough to ﬁnd good models of realworld situations even
using all the insights and results of probability theory as a starting basis.
2
A number of mathematical issues arise with inﬁnite sample spaces, as discussed in the following section. 1.1. PROBABILITY MODELS 3 of equal probabilities seems natural. On closer inspection, there is the following important
diﬀerence.
The model corresponds to a single roll of a die, with a probability deﬁned for each possible
outcome. In a realworld single roll of a die, an outcome k from 1 to 6 occurs, but there is no
observable probability of k. For repeated rolls, however, the realworld relative frequency of
k, i.e., the fraction of rolls for which the output is k, can be compared with the sample value
of the relative frequency of k in a model for repeated rolls. The sample values of the relative
frequency of k in the model resemble the probability of k in a way to be explained later.
It is this relationship through relative frequencies that helps overcome the nonobservable
nature of probabilities in the real world. 1.1.1 The sample space of a probability model An outcome or sample point in a probability model corresponds to the complete result of a
trial of the experiment being modeled. For example, a game of cards is often appropriately
modeled by the arrangement of cards within a shuﬄed 52 card deck, thus giving rise to
a set of 52! outcomes (incredibly detailed, but trivially simple in structure), even though
the entire deck might not be played in one trial of the game. A poker hand with 4 aces is
an event rather than an outcome in this model, since many arrangements of the cards can
give rise to 4 aces in a given hand. The possible outcomes in a probability model (and in
the experiment being modeled) are mutually exclusive and collectively constitute the entire
sample space (space of possible results). An outcome is often called a ﬁnest grain result of
the model in the sense that an outcome ω contains no subsets other than the empty set φ
and the singleton subset {ω }. Thus events typically give only partial information about the
result of the experiment, whereas an outcome fully speciﬁes the result.
In choosing the sample space for a probability model of an experiment, we often omit details
that appear irrelevant for the purpose at hand. Thus in modeling the set of outcomes for
a coin toss as {H, T }, we usually ignore the type of coin, the ini...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.
 Spring '09
 R.Srikant

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