Concrete mathematics : a foundation for computer science

8
Discrete Probability
THE ELEMENT OF CHANCE enters into many of our attempts to understand
the world we live in. A mathematical theory of probability allows us
to calculate the likelihood of complex events if we assume that the events are
governed by appropriate axioms. This theory has significant applications in
all branches of science, and it has strong connections with the techniques we
have studied in previous chapters.
Probabilities are called “discrete” if we can compute the probabilities of
all events by summation instead of by integration. We are getting pretty good
at sums, so it should come as no great surprise that we are ready to apply
our knowledge to some interesting calculations of probabilities and averages.
8.1 DEFINITIONS
(Readers unfamiliar Probability theory starts with the idea of a probability space, which
with probability
theory will, with
high probability,
benefit from a
perusal of Feller’s
is a set fl of all things that can happen in a given problem together with a
rule that assigns a probability Pr(w) to each elementary event w E a. The
probability Pr(w) must be a nonnegative real number, and the condition
classic introduction
to the subject [96].)
x Pr(w) = 1 WEn
(8.1)
must hold in every dimscrete probability space. Thus, each value Pr(w) must lie
in the interval [O . . 11. We speak of Pr as a probability distribution, because
it distributes a total probability of 1 among the events w.
Here’s an example: If we’re rolling a pair of dice, the set 0 of elementary
events is D2 = { � E], � D, . . . , � a}, where
Never say die. is the set of all six ways that a given die can land. Two rolls such as � u
and � n are considered to be distinct; hence this probability space has a
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