Sequence Comparison.pdf

Appendix B
Elementary Probability Theory
This chapter summaries basic background knowledge and establishes the book’s
terminology and notation in probability theory. We give informal definitions for
events, random variables, and probability distributions and list statements without
proof. The reader is referred to consult an elementary probability textbook for the
need of justification.
B.1 Events and Probabilities
Intuitively, an event is something that will or will not occur in a situation depending
on chance. Such a situation is called an experiment. For example, in the experiment
of rolling a die, an event might be that the number turning up is 5. In an alignment
of two DNA sequences, whether the two sequences have the same nucleotide at a
position is an event.
The certain event, denoted by Ω, always occurs. The impossible event, denoted
by φ, never occurs. In different contexts these two events might be described in
different, but essentially identical ways.
Let A be an event. The complementary event of A is the event that A does not
occur and is written Ā.
Let A 1 and A 2 be two events. The event that at least one of A 1 or A 2 occurs
is called the union of A 1 and A 2 and is written A 1 ∪ A 2 . The event that both A 1
and A 2 occur is called the intersection of A 1 and A 2 and is written A 1 ∩ A 2 .We
say A 1 and A 2 are disjoint, ormutually exclusive, if they cannot occur together.
In this case, A 1 ∩ A 2 = φ. These definitions extend to finite number of events in a
natural way. The union of the events A 1 ,A 2 ,...,A n is the event that at least one of
these events occurs and is written A 1 ∪ A 2 ∪···∪A n or ∪ n i=1 A i. The intersection of
the events A 1 ,A 2 ,...,A n is the event that all of these events occur, and is written
A 1 ∩ A 2 ∩···∩A n ,orsimplyA 1 A 2 ···A n .
The probability of an event A is written Pr[A]. Obviously, for the certain event
Ω, Pr[Ω]=1; for the impossible event φ, Pr[φ]=0. For disjoint events A 1 and A 2 ,
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